Solving Quadratic Equations

Before going to learn about solving quadratic equations, let us recall a few facts about quadratic equations. The word "quadratic" is originated from the word "quad" and its meaning is "square". It means the quadratic equation has a variable raised to 2 as the greatest power term. The standard form of a quadratic equation is given by the equation ax² + bx + c = 0, where a ≠ 0. We know that any value(s) of x that satisfies the equation is known as a solution (or) root of the equation and the process of finding the values of x which satisfies the equation ax² + bx + c = 0 is known as solving quadratic equations.

There are different methods used to solve quadratic equations. But most popular method is "solving quadratic equations by factoring". Let us learn all the methods in detail here along with a few solved examples.

1.	How to Solve Quadratic Equations?
2.	Solving Quadratic Equations by Factoring
3.	Solving Quadratic Equations by Completing Square
4.	Solving Quadratic Equations by Graphing
5.	Solving Quadratic Equations by Quadratic Formula
6.	FAQs on Solving Quadratic Equations

How to Solve Quadratic Equations?

Solving quadratic equations means finding a value (or) values of variable which satisfy the equation. The value(s) that satisfy the quadratic equation are known as its roots (or) solutions (or) zeros. Since the degree of the quadratic equation is 2, it can have maximum of 2 roots. For example, one can easily see that x = 1 and x = 2 satisfy the quadratic equation x² - 3x + 2 = 0 (you can substitute each of the values in this equation and verify). Thus, x = 1 and x = 2 are the roots of x² - 3x + 2 = 0. But how to find them if they are not given? There are different ways for solving quadratic equations.

Solving quadratic equations by factoring
Solving quadratic equations by completing the square
Solving quadratic equations by graphing
Solving quadratic equations by quadratic formula

Apart from these methods, there are some other methods that are used only in specific cases (when the quadratic equation has missing terms) as explained below.

Solving Quadratic Equations Missing b

In a quadratic equation ax² + bx + c = 0, if the term with b is missing then the equation becomes ax² + c = 0. This can be solved by taking square root on both sides. The process is explained with examples below.

x² - 4 = 0 ⇒ x² = 4 ⇒ x = ±√4 ⇒ x = ± 2
Thus, the roots of the equation are 2 and -2.
x² + 36 = 0 ⇒ x² = -36 ⇒ x = ±√(-36) ⇒ x = ± 6i
Thus, the roots of the equation are 6i and -6i.
(note that these are imaginary (or) complex numbers).

Solving Quadratic Equations Missing c

In a quadratic equation ax² + bx + c = 0, if the term with c is missing then the equation becomes ax² + bx = 0. To solve this type of equation, we simply factor x out from the left side, set each of the factors to zero, and solve. The process is explained with examples below.

x² - 5x = 0 ⇒ x (x - 5) = 0 ⇒ x = 0; x - 5 = 0 ⇒ x = 0; x = 5
Thus, the roots of the equation are 0 and 5.
x² + 11x = 0 ⇒ x (x + 11) = 0 ⇒ x = 0; x + 11 = 0 ⇒ x = 0; x = -11
Thus, the roots of the equation are 0 and -11.

Now, we will learn the methods of solving the quadratic equations in each of the above mentioned methods.

Solving Quadratic Equations by Factoring

"Solving quadratic equations by factoring" is one of the famous methods used to solve quadratic equations. The step-by-step process of solving quadratic equations by factoring is explained along with an example where we will solve the equation x² - 3x + 2 = 0.

The process of solving quadratic equations by factoring is explained with steps.

Step - 1: Get the equation into standard form. i.e., Get all the terms of to one side (usually to left side) of the equation such that the other side is 0.
The equation x² - 3x + 2 = 0 is already in standard form.
Step - 2: Factor the quadratic expression. If you want to know how to factor a quadratic expression, click here.
Then we get (x - 1) (x - 2) = 0.
Step - 3: By zero product property, set each of the factors to zero.
x - 1 = 0 (or) x - 2 = 0
Step - 4: Solve each of the above equations.
x = 1 (or) x = 2

Thus, the solutions of the quadratic equation x² - 3x + 2 = 0 are 1 and 2. This method is applicable only when the quadratic expression is factorable. If it is NOT factorable, then we can use one of the other methods as explained below.

Solving Quadratic Equations by Completing Square

Completing the square means writing the quadratic expression ax² + bx + c into the form a (x - h)² + k (which is also known as vertex form), where h = -b/2a and 'k' can be obtained by substituting x = h in ax² + bx + c. The step-by-step process of solving the quadratic equations by completing the square is explained along with an example where we are going to find the solutions of the equation 2x² + 8x = -3.

The process of solving quadratic equations by completing square is explained with steps.

Step - 1: Get the equation into standard form.
Adding 3 on both sides, we get 2x² + 8x + 3 = 0.
Step - 2: Complete the square on the left side.
Then we get 2 (x + 2)² - 5 = 0. If you want to know how to complete the square, click here.
Step - 3: Solve it for x (We will have to take square root on both sides along the way).
Adding 5 on both sides,
2 (x + 2)² = 5
Dividing both sides by 2,
(x + 2)² = 5/2
Taking square root on both sides,
x + 2 = √(5/2) = √5/√2 · √2/√2 = √10/2
Subtracting 2 from both sides,
x = -2 ± (√10/2) = (-4 ± √10) / 2

Thus, the roots of the quadratic equation 2x² + 8x = -3 are (-4 + √10)/2 and (-4 - √10)/2.

Solving Quadratic Equations by Graphing

For solving the quadratic equations by graphing, we first have to graph the quadratic expression (when the equation is in the standard form) either manually or by using a graphing calculator. Then the x-intercept(s) of the graph (the point(s) where the graph cuts the x-axis) are nothing but the roots of the quadratic equation. Here are the steps to solve quadratic equations by graphing which are explained along with an example where we are going to solve the equation 3x² + 5 = 11x.

The process of solving quadratic equations by graphing is explained with steps.

Step - 1: Get into the standard form.
Subtracting 11x from both sides, 3x²- 11x + 5 = 0.
Step - 2: Graph the quadratic expression (which is on the left side).
Graph the quadratic function y = 3x²- 11x + 5 either manually or using a graphing calculator (GDC).
You can see the graph in the step below.
Step - 3: Identify the x-intercepts.
Step - 4: The x-coordinates of the x-intercepts are nothing but the roots of the quadratic equation.
Thus, the solutions of the quadratic equation 3x² + 5 = 11x are 0.532 and 3.135.

By seeing the above example, we can see that the graphing method of solving quadratic equations may not give the exact solutions (i.e., it gives only the decimal approximations of the roots if they are irrational). i.e., if we solve the same equation using completing the square, we get x = (11 + √61) / 6 and x = (11 - √61) / 6. But we cannot get these exact roots by the graphing method.

What if the graph does not intersect the x-axis at all? It means that the quadratic equation has two complex roots. i.e., the graphing method is NOT helpful to find the roots if they are complex numbers. We can use the quadratic formula (which is explained in the next section) to find any type of roots.

Solving Quadratic Equations by Quadratic Formula

As we have already seen, the previous methods for solving the quadratic equations have some limitations such as the factoring method is useful only when the quadratic expression is factorable, the graphing method is useful only when the quadratic equation has real roots, etc. But solving quadratic equations by quadratic formula overcomes all these limitations and is useful to solve any type of quadratic equation. Here is the step-by-step explanation of solving quadratic equations by quadratic formula along with an example where we will be finding the solutions of the quadratic equation 2x² = 3x - 5.

The process of solving quadratic equations by quadratic formula is explained with steps.

Step - 1: Get into the standard form.
Then the above equation becomes 2x² - 3x + 5 = 0.
Step - 2: Compare the equation with ax² + bx + c = 0 and find the values of a, b, and c.
Then we get a = 2, b = -3. and c = 5.
Step - 3: Substitute the values into the quadratic formula which says x = [-b ± √(b² - 4ac)] / (2a). Then we get
x = [-(-3) ± √((-3)² - 4(2)(5))] / (2(2))
Step - 4: Simplify.
x = [ 3 ± √(9 - 40) ] / 4
= [ 3 ± √(-31) ] / 4
= [ 3 ± i√(31) ] / 4

Thus, the roots of the quadratic equation 2x² = 3x - 5 are [ 3 + i√(31) ] / 4 and [ 3 - i√(31) ] / 4. In the quardratic formula, the expression b² - 4ac is called the discriminant (that is denoted by D). i.e., D = b² - 4ac. This is used to determine the nature of roots of the quadratic equation.

Nature of Roots Using Discriminant

If D > 0, then the equation ax² + bx + c = 0 has two real and distinct roots.
If D = 0, then the equation ax² + bx + c = 0 has only one real root.
If D < 0, then the equation ax² + bx + c = 0 has two distinct complex roots.

Thus, using the discriminant, we can find the number of solutions of quadratic equations without actually solving it.

Important Notes on Solving Quadratic Equations:

The factoring method cannot be applied when the quadratic expression is NOT factorable.
The graphing method cannot give the complex roots and also it cannot give the exact roots in case the quadratic equation has irrational roots.
Completing the square method and quadratic formula method can be applied to solve any type of quadratic equation.
The roots of the quadratic equation are also known as "solutions" or "zeros".
For any quadratic equation ax² + bx + c = 0,
the sum of the roots = -b/a
the product of the roots = c/a.

Topics Related to Quadratic Equations:

Examples on Solving Quadratic Equations

Example 1: The length of a park is 5 ft less than twice its width. Find the dimensions of the park if its area is 250 square feet.

Solution:

Let the width of the park be x ft.

Then the length of the park = (2x - 5) ft.

Its area = 250 ft²

length × width = 250

(2x - 5) x = 250

2x² - 5x - 250 = 0

Hence, this is a word problem related to solving quadratic equations. Let us solve this quadratic equation by factoring.

Here a = 2, b = -5 and c = -250.

ac = 2(-250) = -500.

Two numbers whose sum is -5 and whose product is -500 are -25 and 20. So we split the middle term using these two numbers.

2x² - 25x + 20x - 250 = 0

x (2x - 25) + 10 (2x - 25) = 0

(2x - 25) (x + 10) = 0

2x - 25 = 0 (or) x + 10 = 0

x = 25/2 = 12.5 (or) x = -10

x = 12.5 as x cannot be negative.

So width = 12.5 ft and length = (2x - 5) ft = 2(12.5) - 5 = 20 ft.

Answer: The dimensions of the park are 20 ft × 12.5 ft.
Example 2: If twice the difference of a number and 6 is equal to -2 times its square, then find the number(s).

Solution:

Let the required number be x. Then

2(x - 6) = -2x²

Let us solve this quadratic equation by factoring. For this, we have to convert this into standard form. Then

2x² + 2x - 12 = 0

Here a = 2, b = 2 and c = -12.

ac = 2(-12) = -24.

Two numbers whose sum is 2 and whose product is -24 are 6 and -4. So we split the middle term using these two numbers.

2x² + 6x - 4x - 12 = 0

2x (x + 3) - 4 (x + 3) = 0

(x + 3) (2x - 4) = 0

x + 3 = 0, 2x - 4 = 0

x = -3, x = 2

Answer: The required numbers are -3 and 2.
Example 3: The product of two positive consecutive numbers is 156. Find the two numbers.

Solution:

Let us assume that the two consecutive numbers be x and x + 1. Then

x (x + 1) = 156

x² + x - 156 = 0

Let us solve this quadratic equation by factoring.

Here a = 1, b = 1 and c = -156.

ac = 1(-156) = -156.

Two numbers whose sum is 1 and whose product is -156 are 13 and -12. So we split the middle term using these two numbers.

x² + 13x - 12x - 156 = 0

x (x + 13) - 12 (x + 13) = 0

(x + 13) (x - 12) = 0

x + 13 = 0, x - 12 = 0

x = -13 (or) x = 12

Since x is positive (given), x cannot be -13. So x = 12.

Answer: The required consecutive numbers are 12 and 13 (12 + 1).

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Practice Questions on Solving Quadratic Equations

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FAQs on Solving Quadratic Equations

What is the Meaning of Solving Quadratic Equations?

Solving quadratic equations means finding its solutions or roots. i.e., it is the process of finding the values of the variable that satisfy the equation.

How to Solve Quadratic Equations?

There are different ways to solve quadratic equations. But the most popular ways are "solving quadratic equations by factoring" and "solving quadratic equations by quadratic formula".

What are 4 ways to Solve a Quadratic Equation?

There are 4 ways for solving quadratic equations.

by factoring
by completing square
by graphing
by quadratic formula

How to Solve Quadratic Equations by Quadratic Formula?

The solutions of a quadratic equation ax² + bx + c = 0 are given by the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). So to solve a quadratic equation using quadratic formula, just get the equation into standard form ax² + bx + c = 0, and apply the quadratic formula.

What is the Easiest Way For Solving Quadratic Equations?

The easiest way of solving quadratic equations is the factoring method. But not always quadratic expressions are factorable. In that case, we can either use the quadratic formula or use completing square method.

What are the Steps in Solving Quadratic Equations by Completing Square?

To solve the quadratic equation ax² + bx + c = 0 by completing square, convert ax² + bx + c into the form a (x - h)² + k where h = -b/2a and k is obtained by substituting x = h in ax² + bx + c. Then we can easily solve a (x - h)² + k = 0 by isolating x. In this process, we will have to take the square root on both sides.

How do You Know Which Method to Use When Solving Quadratic Equations?

We can solve the quadratic equations of any type using completing the square or the quadratic formula. But if the quadratic expression is factorable, then the factoring method is the easiest to apply. We can solve it by graphing method also, but it gives only approximated real roots (i.e., complex roots cannot be found in this method).

How to Solve Quadratic Equations by Factoring?

For solving the quadratic equations by factoring, first convert it into the standard form (ax² + bx + c = 0). Then factorize the left side part part using the techniques of factorizing quadratic expressions, set each of the factors to zero that results in two linear equations, and finally solve the linear equations.

How is the Factored Form Helpful in Solving Quadratic Equations?

If the quadratic expression that is in the standard form of quadratic expression in it is factorable, then we can just set each factor to zero, and solve them. The solutions are nothing but the roots of the quadratic equation.

How to Find the Roots of Quadratic Equations?

The roots of the quadratic equation ax² + bx + c = 0 can be found by using the quadratic formula that says x = [-b ± √(b² - 4ac)] / (2a). Also, we can solve them by completing square (or) factoring method (only when they are factorable).

What are the Steps in Solving Quadratic Equations by Graphing?

To solve the quadratic equations by graphing, first get into standard form ax² + bx + c = 0. Then graph the quadratic expression ax² + bx + c. Find where the graph is intersecting the x-axis. The x-coordinate of the x-intercept(s) are nothing but the solutions of the quadratic equation.

Which Method is Best For Solving Quadratic Equations?

The best method to solve quadratic equations is factoring. But when factoring is not possible, we solve them using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). If you have a graphing calculator, then graphing method would be the easiest to find the decimal approximation of roots (we cannot find exact roots though).

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