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 satisfy the equation ax² + bx + c = 0 is known as solving quadratic equations.

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

Solving quadratic equations means finding a value (or) values of variable which satisfy the equation. The value(s) that satisfy the quadratic equation is known as its roots (or) solutions (or) zeros. Since the degree of the quadratic equation is 2, it can have a 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 of 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 + 49 = 0 ⇒ x² = -49 ⇒ x = ±√(-49) ⇒ x = ± 7i
  Thus, the roots of the equation are 7i and -7i.
  (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 + 21x = 0 ⇒ x (x + 21) = 0 ⇒ x = 0; x + 21 = 0 ⇒ x = 0; x = -21
  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 quadratics 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.

• 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.
• Step - 2: Factor the quadratic expression. If you want to know how to factor a quadratic expression, click here.
• Step - 3: By zero product property, set each of the factors to zero.
• Step - 4: Solve each of the above equations.

Example: Solve the quadratic equation x² - 3x + 2 = 0 by factoring it.

Solution:

Factoring the left side part, we get (x - 1) (x - 2) = 0.

Then x - 1 = 0 (or) x - 2 = 0

which gives 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. Similar to quadratic equations we have solutions for linear equations, which are used to solve linear programming problems.

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.

• Step - 1: Get the equation into standard form.
• Step - 2: Complete the square on the left side. 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).

Example: Solve 2x² + 8x = -3 by completing the square.

Solution:

The given equation in the standard form is 2x² + 8x + 3 = 0. Completing the square on the left side, we get 2 (x + 2)² - 5 = 0. Now solving it for x,

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.

For solving the quadratics 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.

• Step - 1: Get into the standard form.
• Step - 2: Graph the quadratic expression (which is on the left side).
• Step - 3: Identify the x-intercepts.
• Step - 4: The x-coordinates of the x-intercepts are nothing but the roots of the quadratic equation.

Example: Solve the quadratic 3x² + 5 = 11x by graphing.

Solution:

Converting the given equation into the standard form, subtracting 11x from both sides, 3x² - 11x + 5 = 0. Now, graph the quadratic expression Graph the quadratic function y = 3x² - 11x + 5 either manually or using a graphing calculator (GDC) and identify the x-intercepts. Its graph is

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.

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 quadratics by quadratic formula.

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

Example: Solve the quadratic equation 2x² = 3x - 5 by the quadratic formula.

Solution:

The above equation in standard form is 2x² - 3x + 5 = 0.

Comparing the equation with ax² + bx + c = 0, we get a = 2, b = -3. and c = 5.

Substitute the values into the quadratic formula

x = [-(-3) ± √((-3)² - 4(2)(5))] / (2(2))
= [ 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.

☛Related Topics:

• Solving Quadratic Equations by Quadratic Formula Calculator
• Solving Quadratic Equations by Completing Square Calculator
• Roots of Quadratic Equation Calculator
• Solving Quadratic Equations by Factoring Calculator

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.

What are the Most Popular Ways of Solving Quadratic Equations?

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

What are the Steps in Solving Quadratic Equations by Graphing?

To solve quadratics 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.

What are 4 ways to Solving Quadratics?

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.

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).

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 the square method.

What are the Steps in Solving Quadratics by Completing Square?

To solve the quadratic equation ax² + bx + c = 0 by completing square, convert ax² + bx + c into the vertex 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 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 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).

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).