How do you solve quadratic equations by using the quadratic formula

Quadratic equations are second-degree algebraic expressions and are of the form ax2 + bx + c = 0. The word "Quadratic" is derived from the word "Quad" which means square. In other words, a quadratic equation is an “equation of degree 2.” There are many scenarios where a quadratic equation is used. Did you know that when a rocket is launched, its path is described by a quadratic equation? Further, a quadratic equation has numerous applications in physics, engineering, astronomy, etc.

The quadratic equations are second-degree equations in x that have a maximum of two answers for x. These two answers for x are also called the roots of the quadratic equations and are designated as (α, β). We shall learn more about the roots of a quadratic equation in the below content.

What is Quadratic Equation?

A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term. The first condition for an equation to be a quadratic equation is the coefficient of x2 is a non-zero term(a ≠ 0). For writing a quadratic equation in standard form, the x2 term is written first, followed by the x term, and finally, the constant term is written. The numeric values of a, b, c are generally not written as fractions or decimals but are written as integral values.

Further in real math problems the quadratic equations are presented in different forms: (x - 1)(x + 2) = 0, -x2 = -3x + 1, 5x(x + 3) = 12x, x3 = x(x2 + x - 3). All of these equations need to be transformed into standard form of the quadratic equation before performing further operations.

Roots of a Quadratic Equation

The roots of a quadratic equation are the two values of x, which are obtained by solving the quadratic equation. These roots of the quadratic equation are also called the zeros of the equation. For example, the roots of the equation x2 - 3x - 4 = 0 are x = -1 and x = 4 because each of them satisfy the equation. i.e.,

• At x = -1, (-1)2 - 3(-1) - 4 = 1 + 3 - 4 = 0
• At x = 4, (4)2 - 3(4) - 4 = 16 - 12 - 4 = 0

There are various methods to find the roots of a quadratic equation. The usage of quadratic formula is one of them.

Quadratic Formula

Quadratic Formula is the simplest way to find the roots of a quadratic equation. There are certain quadratic equations that cannot be easily factorized, and here we can conveniently use this quadratic formula to find the roots in the quickest possible way. The roots of the quadratic equation further help to find the sum of the roots and the product of the roots of the quadratic equation. The two roots in the quadratic formula are presented as a single expression. The positive sign and the negative sign can be alternatively used to obtain the two distinct roots of the equation.

Quadratic Formula: The roots of a quadratic equation ax2 + bx + c = 0 are given by x = [-b ± √(b2 - 4ac)]/2a.

Example: Let us find the roots of the same equation that was mentioned in the earlier section x2 - 3x - 4 = 0 using the quadratic formula.

a = 1, b = -3, and c = -4.

x = [-b ± √(b2 - 4ac)]/2a
= [-(-3) ± √((-3)2 - 4(1)(-4))]/2(1)
= [3 ± √25] / 2
= [3 ± 5] / 2
= (3 + 5)/2 or (3 - 5)/2
= 8/2 or -2/2
= 4 or -1 are the roots.

Quadratic Formula Proof

Consider an arbitrary quadratic equation: ax2 + bx + c = 0, a ≠ 0

To determine the roots of this equation, we proceed as follows:

ax2 + bx = -c ⇒ x2 + bx/a = -c/a

Now, we express the left-hand side as a perfect square, by introducing a new term (b/2a)2 on both sides:

x2+ bx/a + (b/2a)2 = -c/a + (b/2a)2

The left hand side is now a perfect square:

(x + b/2a)2 = -c/a + b2/4a2 ⇒ (x + b/2a)2 = (b2 - 4ac)/4a2

This is good for us, because now we can take square roots to obtain:

x + b/2a = ±√(b2 - 4ac)/2a

x = (-b ± √(b2 - 4ac))/2a

Thus, by completing the squares, we were able to isolate x and obtain the two roots of the equation.

Nature of Roots of the Quadratic Equation

The roots of a quadratic equation are usually represented to by the symbols alpha (α), and beta (β). Here we shall learn more about how to find the nature of roots of a quadratic equation without actually finding the roots of the equation. And also check out the formulas to find the sum and the product of the roots of the equation.

The nature of roots of a quadratic equation can be found without actually finding the roots (α, β) of the equation. This is possible by taking the discriminant value, which is part of the formula to solve the quadratic equation. The value b2 - 4ac is called the discriminant of a quadratic equation and is designated as 'D'. Based on the discriminant value the nature of the roots of the quadratic equation can be predicted.

Discriminant: D = b2 - 4ac

• D > 0, the roots are real and distinct
• D = 0, the roots are real and equal.
• D < 0, the roots do not exist or the roots are imaginary.

Sum and Product of Roots of Quadratic Equation

The coefficient of x2, x term, and the constant term of the quadratic equation ax2+ bx + c = 0 are useful in determining the sum and product of the roots of the quadratic equation. The sum and product of roots of a quadratic equation can be directly calculated from the equation, without actually finding the roots of the quadratic equation. The sum of the roots of the quadratic equation is equal to the negative of the coefficient of x divided by the coefficient of x2. The product of the root of the equation is equal to the constant term divided by the coefficient of the x2. For a quadratic equation ax2 + bx + c = 0, the sum and product of the roots are as follows.

• Sum of the Roots: α + β = -b/a = - Coefficient of x/ Coefficient of x2
• Product of the Roots: αβ = c/a = Constant term/ Coefficient of x2

The quadratic equation can also be formed for the given roots of the equation. If α, β, are the roots of the quadratic equation, then the quadratic equation is as follows.

x2 - (α + β)x + αβ = 0

Formulas Related to Quadratic Equations

The following list of important formulas is helpful to solve quadratic equations.

• The quadratic equation in its standard form is ax2 + bx + c = 0
• The discriminant of the quadratic equation is D = b2 - 4ac
• For D > 0 the roots are real and distinct.
• For D = 0 the roots are real and equal.
• For D < 0 the real roots do not exist, or the roots are imaginary.
• The formula to find the roots of the quadratic equation is x = [-b ± √(b2 - 4ac)]/2a.
• The sum of the roots of a quadratic equation is α + β = -b/a.
• The product of the Root of the quadratic equation is αβ = c/a.
• The quadratic equation whose roots are α, β, is x2 - (α + β)x + αβ = 0.
• The condition for the quadratic equations a1x2 + b1x + c1 = 0, and a2x2 + b2x + c2 = 0 having the same roots is (a1b2 - a2b1) (b1c2 - b2c1) = (a2c1 - a1c2)2.
• When a > 0, the quadratic expression f(x) = ax2 + bx + c has a minimum value at x = -b/2a.
• When a < 0, the quadratic expression f(x) = ax2 + bx + c has a maximum value at x = -b/2a.
• The domain of any quadratic function is the set of all real numbers.
• For a > 0, the range of the quadratic function f(x) = ax2 + bx + c is [-(b2 - 4ac)/4a, ∞)
• For a < 0, the range of the quadratic function f(x) = ax2 + bx + c is : (-∞, -(b2 - 4ac)/4a]

Methods to Solve Quadratic Equations

A quadratic equation can be solved to obtain two values of x or the two roots of the equation. There are four different methods to find the roots of the quadratic equation. The four methods of solving the quadratic equations are as follows.

• Factorizing of Quadratic Equation
• Using quadratic formula (which we have seen already)
• Method of Completing the Square
• Graphing Method to Find the Roots

Let us look in detail at each of the above methods to understand how to use these methods, their applications, and their uses.

Factorization of Quadratic Equation

Factorization of quadratic equation follows a sequence of steps. For a general form of the quadratic equation ax2 + bx + c = 0, we need to first split the middle term into two terms, such that the product of the terms is equal to the constant term. Further, we can take the common terms from the available term, to finally obtain the required factors. For understanding factorization, the general form of the quadratic equation can be presented as follows.

• x2 + (a + b)x + ab = 0
• x2 + ax + bx + ab = 0
• x(x + a) + b(x + a)
• (x + a)(x + b) = 0

Let us understand factorization through the below example.

• x2 + 5x + 6 = 0
• x2 + 2x + 3x + 6 = 0
• x(x + 2) + 3(x + 2) = 0
• (x + 2)(x + 3) = 0

Thus the two obtained factors of the quadratic equation are (x + 2) and (x + 3). To find its roots, just set each factor to zero and solve for x. i.e., x + 2 = 0 and x + 3 = 0 which gives x = -2 and x = -3. Thus, x = -2 and x = -3 are the roots of x2 + 5x + 6 = 0.

Further, there is another important method of solving a quadratic equation. The method of completing the square for a quadratic equation is also useful to find the roots of the equation.

Method of Completing the Square

The method of completing the square in a quadratic equation is to algebraically square and simplify, to obtain the required roots of the equation. Consider a quadratic equation ax2 + bx + c = 0, a ≠ 0. To determine the roots of this equation, we simplify it as follows:

• ax2 + bx + c = 0
• ax2 + bx = -c
• x2 + bx/a = -c/a

Now, we express the left hand side as a perfect square, by introducing a new term (b/2a)2 on both sides:

• x2 + bx/a + (b/2a)2 = -c/a + (b/2a)2
• (x + b/2a)2 = -c/a + b2/4a2
• (x + b/2a)2 = (b2 - 4ac)/4a2
• x + b/2a = +√(b2- 4ac)/2a
• x = - b/2a +√(b2- 4ac)/2a

Now with this method of completing the square, we could consolidate the value for the roots of the equation. Further on simplification and taking the square root, the two possible roots of the quadratic equation are, x = (-b + √(b2- 4ac))/2a. Here the '+' sign gives one root and the '-' sign gives another root of the quadratic equation. Generally, this detailed method is avoided, and only the quadratic formula is used to obtain the required roots.

Graphing a Quadratic Equation

The graph of the quadratic equation ax2 + bx + c = 0 can be obtained by representing the quadratic equation as a function y = ax2 + bx + c. Further on solving and substituting values for x, we can obtain values of y, we can obtain numerous points. These points can be presented in the coordinate axis to obtain a parabola-shaped graph for the quadratic equation. For detailed information about graphing a quadratic function, click here.

The point(s) where the graph cuts the horizontal x-axis (typically the x-intercepts) is the solution of the quadratic equation. These points can also be algebraically obtained by equalizing the y value to 0 in the function y = ax2 + bx + c and solving for x.

Quadratic Equations Having Common Roots

Let the two quadratic equations having common roots are a1x2 + b1x + c1 = 0, and a2x2 + b2x + c2 = 0. Let us solve these two equations to find the conditions for which these equations have a common root. The two equations are solved for x2 and x respectively.

(x2)(b1c2 - b2c1) = (-x)/(a1c2 - a2c1) = 1/(a1b2 - a2b1)

x2 = (b1c2 - b2c1) / (a1b2 - a2b1)

x = (a2c1 - a1c2) / (a1b2 - a2b1)

Hence, by simplifying the above two expressions we have the following condition for the two equations having the common root.

(a1b2 - a2b1) (b1c2 - b2c1) = (a2c1 - a1c2)2

Maximum and Minimum Value of Quadratic Expression

The maximum and minimum value for the quad equation F(x) = ax2 + bx + c = 0 can be observed in the below graphs. For positive values of a (a > 0), the quadratic expression has a minimum value at x = -b/2a, and for negative value of a (a < 0), the quadratic expression has a maximum value at x = -b/2a. x = -b/2a is the x-coordinate of the vertex of the parabola.

The maximum and minimum values of the quadratic expressions are of further help to find the range of the quadratic expression: The range of the quadratic expressions also depends on the value of a. For positive values of a( a > 0), the range is [ F(-b/2a), ∞), and for negative values of a ( a < 0), the range is (-∞, F(-b/2a)].

• For a > 0, Range: [ f(-b/2a), ∞)
• For a < 0, Range: (-∞, f(-b/2a)]

Note that the domain of a quadratic function is the set of all real numbers, i.e., (-∞, ∞).

Tips and Tricks on Quadratic Equation:

Some of the below-given tips and tricks on quadratic equations are helpful to more easily solve quadratic equations.

• The quadratic equations are generally solved through factorization. But in instances when it cannot be solved by factorization, the quadratic formula is used.
• The roots of a quadratic equation are also called the zeroes of the equation.
• For quadratic equations having negative discriminant values, the roots are represented by complex numbers.
• The sum and product of roots of a quadratic equation can be used to find higher algebraic expressions involving these roots.

☛Related Topics:

• Roots Calculator
• Roots of Quadratic Equation Calculator

FAQs on Quadratic Equation

What is the Definition of a Quadratic Equation?

A quadratic equation in math is a second-degree equation of the form ax2 + bx + c = 0. Here a, b, are the coefficients, c is the constant term, and x is the variable. Since the variable x is of the second degree, there are two roots or answers for this quadratic equation. The roots of the quadratic equation can be found by either solving by factorizing or through the use of the quadratic formula.

What is the Quadratic Formula?

The quadratic formula to solve a quadratic equation ax2 + bx + c = 0 is x = [-b ± √(b2 - 4ac)]/2a. Here we obtain the two values of x, by applying the plus and minus symbol in this formula. Hence the two possible values of x are [-b + √(b2 - 4ac)]/2a, and [-b - √(b2 - 4ac)]/2a.

How To Find Nature of Roots of Quadratic Equation?

The discriminant is helpful to predict the nature of the roots of the quadratic equation. The discriminant of a quadratic equation ax2 + bx + c = 0 is b2 - 4ac. The discriminant is referred as D = b2 - 4ac. If D > 0 the roots are real and distinct, for D = 0 the roots are equal, and for D < 0 the roots are imaginary complex numbers.

How to Apply Quadratic Formula?

The values of a, b, and c are substituted in the quadratic formula x = [-b ± √(b2 - 4ac)]/2a, to obtain the two roots of the quadratic equation. Do not forget to apply + and - signs separately.

What is Determinant in Quadratic Formula?

The value b2 - 4ac is called the discriminant and is designated as D. The discriminant is part of the quadratic formula. The discriminants help us to find the nature of the roots of the quadratic equation, without actually finding the roots of the quadratic equation.

What are Some Real-Life Applications of Quadratic Equations?

Quadratic equations are used to find the zeroes of the parabola and its axis of symmetry. There are many real-world applications of quadratic equations. For instance, it can be used in running time problems to evaluate the speed, distance or time while traveling by car, train or plane. Quadratic equations describe the relationship between quantity and the price of a commodity. Similarly, demand and cost calculations are also considered quadratic equation problems. It can also be noted that a satellite dish or a reflecting telescope has a shape that is defined by a quadratic equation.

How are Quadratic Equations Different From Linear Equations?

A linear degree is an equation of a single degree and one variable, and a quadratic equation is an equation in two degrees and a single variable. A linear equation is of the the form ax + b = 0 and a quadratic equation is of the form ax2 + bx + c = 0. A linear equation has a single root and a quadratic equation has two roots or two answers. Also, a quadratic equation is a product of two linear equations.

How to Simplify a Quadratic Equation?

The first step in the process of simplifying a quadratic equation is to transform it into the standard form ax2 + bx + c = 0. Further, it can be simplified by finding its factors through the process of factorization. Also for an equation for which it is difficult to factorize, it is solved by using the formula. Additionally, there are a few other ways of simplifying a quadratic equation.

What Are the 4 Ways To Solve A Quadratic Equation?

The four ways of solving a quadratic equation are as follows.

• Factorizing method
• Formula Method
• Method of Completing Squares
• Graphing Method

How Do you Solve a Quad Equation By Factoring?

The quad equation can be solved by factorization through a sequence of three steps. First split the middle term, such that the product of the split terms is equal to the product of the first and the last terms. Let us assume the quadratic equation is of the form x2 + (a + b)x + ab = 0, and it can be split as x2 + ax + bx + ab = 0. As a second step, take the common term from the first two and the last two terms. x(x + a) + b(x + a) = 0, (x + a)(x + b) = 0. Finally, equalize each of the factors to zero and obtain the x values. x + a = 0 and x + b = 0, and hence we can obtain x = -a and x = -b

How to Solve a Quadratic Equation by Completing the Square?

The quadratic equation is solved by the method of completing the square and it uses the formula (a + b)^2 = a2 + 2ab + b2 (or) (a - b)^2 = a2 - 2ab + b2.

How to Find the Value of the Discriminant?

The value of the discriminant in a quadratic equation can be found from the variables and constant terms of the standard form of the quadratic equation ax2 + bx + c = 0. The value of the discriminant is D = b2 - 4ac, and it helps to predict the nature of roots of the quadratic equation, without actually finding the roots of the equation.

How Do You Solve Quadratic Equations With Graphing?

The quadratic equation can be solved similarly to a linear equal by graphing. Let us take the quadratic equation ax2 + bx + c = 0 as y = ax2 + bx + c . Here we take the set of values of x and y and plot the graph. The two points where this graph meets the x-axis, are the solutions of this quadratic equation.

How Important Is the Discriminant in Determining the Nature of Roots of Quadratic Equation?

The discriminant is very much needed to easily find the nature of the roots of the quadratic equation. Without the discriminant, finding the nature of the roots of the equation is a long process, as we first need to solve the equation to find both the roots. Hence the discriminant is an important and needed quantity, which helps to easily find the nature of the roots of the quadratic equation.

Where Can I Find Quadratic Equation Solver?

To get the quadratic equation solver, click here. Here, we can enter the values of a, b, and c for the quadratic equation ax2 + bx + c = 0, then it will give you the roots along with a step-by-step procedure.

When Do Quad Equations Have Equal Roots?

The given quad equation has equal roots if the discriminant is equal to zero. For a quadratic equation of the form ax2 + bx + c = 0 the discriminant is D = b2 - 4ac = 0. Here both are roots are equal and each has a value of x = -b/2a.

What is the Use of Discriminants in Quadratic Formula?

The discriminant (D = b2 - 4ac) is useful to predict the nature of the roots of the quadratic equation. For D > 0, the roots are real and distinct, for D = 0 the roots are real and equal, and for D < 0, the roots do not exist or the roots are imaginary complex numbers. With the help of this discriminant and with the least calculations, we can find the nature of the roots of the quadratic equation.

How Many Roots does Quadratic Equation Have?

It is a second-degree equation in x, and hence two roots are obtained. We can obtain these roots of a quadratic equation using the quadratic formula. One root can be obtained using the positive sign and we can get another root by applying the negative sign in the formula.

How do you Solve a Quadratic Equation without Using the Quadratic Formula?

There are two alternative methods to the quadratic formula. One method is to solve the quadratic equation through factorization, and another method is by completing the squares. In total there are three methods to find the roots of a quadratic equation.

How to Derive Quadratic Formula?

The algebra formula (a + b)2 = a2 + 2ab + b2 is used to solve the quadratic equation and derive the quadratic formula. This algebraic formula is used to manipulate the quadratic equation and derive the quadratic formula to find the roots of the equation.

What are the 4 steps we used to solve using a quadratic formula?

What are the advantages of using the quadratic formula in solving quadratic equations?