Solving Quadratic Equations by Factoring

 

Learning Objective(s)

·         Solve quadratic equations using factoring techniques and express the solution(s) as a set.

 

 

When any polynomial is set equal to a certain value (whether an integer or another polynomial), the result is an equation. An equation that can be written in the form ax² + bx + c = 0 is called a quadratic equation. We can solve these quadratic equations by using the rules of algebra, applying factoring techniques where necessary, and by using the Zero Product Property.

 

 

The Zero Product Property states (in algebraic terms, of course!) something that we all have known for a long time: if the product of two numbers is 0, then at least one of the factors is 0.

 

Zero Product Property   If ab = 0, then either a = 0 or b = 0, or both a and b are 0.

 

This property may seem fairly obvious, but it has big implications for how we solve quadratic equations: it means that if we have a factored polynomial equal to 0, we can be sure that at least one of the factors is also 0. We can use this method to identify solutions for the equation.

 

But we’re getting ahead of ourselves here—let’s start with an example of a quadratic equation and think about how we can solve it. The equation 5a² + 15a = 0 is a quadratic equation because it can be written as 5a² + 15a + 0 = 0, which fits the form ax² + bx + c = 0, with c = 0.

 

 

At this point we have fully factored the left side of the equation. If we were just factoring the expression, we could stop here, but remember that we are solving the equation for a.

 

Here’s where we use the Zero Product Property. Since the entire expression is equal to zero, we know that at least one of the terms, 5a or (a + 3), has to be equal to zero. Let’s continue to solve this problem by setting each term equal to zero and solving the

equations.

 

 

We end up with two possible values for a: 0 and -3. (These values are also called the roots of an equation.) To check our answers, we can substitute both values directly into our original equation and see if we get a true sentence for each.

 

 

Substituting these values into the original equation produces two correct statements, so we know that our values for a are correct! This quadratic equation, 5a² + 15a = 0, has two roots: 0 and -3.

 

We can use the same Zero Product Property to solve quadratic equations in the form ax² + bx + c = 0. First we factor the expression, and then we will solve for the roots.

 

 

 

The solution to this equation, then, is r = 2 or r = 3, as both of these values for r will result in a true statement. (Skeptical? Substitute values of 2 and 3 in for r in the original equation. We'll wait.)

 

Solve for h:   h(2h + 5) = 0   A) h = 0   B) h = 2 or 5   C) h = 0 or 2.5   D) h = 0 or -2.5   Show/Hide Answer A) Incorrect. While h = 0 does make the equation true (since the first factor is h), it’s possible that the second factor could be 0 as well. This happens when h = -2.5, so the correct answer is h = 0 or -2.5.   B) Incorrect.  The Zero Product Property says if h(2h + 5) = 0 then either h = 0 or 2h + 5 = 0. This happens when h = 0 or -2.5.   C) Incorrect. While h = 0 does make the equation true (since the first factor is h), the second factor is 0 when h = -2.5, not 2.5. The correct answer is h = 0 or -2.5.   D) Correct. To find the roots of this equation, apply the Zero Product Property and set each factor, h and (2h + 5), equal to 0. Then solve those equations for h. Both answers are possible solutions.

 

 

 

When we use the Zero Product Property to solve a quadratic equation, we need to make sure that the equation itself has been set to zero. Sometimes this will require moving terms around in order to make one side zero.

 

As an example, think about the equation 12x² + 11x + 2 = 7. We could factor the trinomial on the left side of the equation as it is, but we would be left with the equation (4x + 1)(3x + 2) = 7. And this is as far as we could get! All this new equation tells us is that two factors, (4x + 1) and (3x + 2), when multiplied together, equal 7. Setting each factor equal to 7 and then solving the equation does not help us either; we are not looking for factors that are 7; we are looking for two factors that, when multiplied together, equal 7. Put another way, we cannot use the Zero Product Property when there is not a zero on one side of the quadratic equation!

 

What’s the solution here? In order to have a zero on one side of the equation, we have to subtract 7 from both sides. This means our quadratic equation goes from 12x² + 11x + 2 = 7 to 12x² + 11x – 5 = 0. We can factor the trinomial on the left and then use the Zero Product Property to find values for x.

 

The example below shows how to solve a quadratic equation where neither side is originally equal to zero. (Note that the factoring sequence has been shortened.)

 

 

 

Sometimes we may factor quadratic equations that end up looking like this: 8(x + 3)(x + 2) = 0. We know how to apply the Zero Product Property to the factors (x + 3) and (x + 2), but what do we do with the leading coefficient of 8? Can we apply the Zero Product Property to an integer?

 

In this situation, we have three factors: 8, x + 3, and x + 2. The Zero Product Rule tells us that if the product of the three factors, 8(x + 3)(x + 2), is going to equal the zero on the right side of the equation, the only way that can happen is if at least one of the three factors on the left side is zero. So let’s check each of them:

 

The factor 8 is never going to be equal to zero, so we can effectively ignore it as one of the choices.

 

The factor x + 3 might equal zero, and we find that when x = -3 then it does.

 

The factor x + 2 might equal zero, and we find that when x = -2 then it does.

 

So our solutions to the original equation are x = -3  or  x = -2, the factor of 8 did not contribute a third solution.

 

Solve for m.   2m2 + 10m = 48   A) m =  3 or -8   B) m = -3 or 8   C) m = 0 or -5   D) m = 0 or 5   Show/Hide Answer A) Correct. To find the roots of this equation, use this procedure: The original equation has 48 on the right. To make this side equal to 0, subtract 48 from both sides: 2m2 + 10m – 48 = 0 Factor out the common factor, 2: 2(m2 + 5m – 24) = 0 Rewrite 5m for easier grouping and factoring: 2(m2 + 8m – 3m – 24) = 0 Use Distributive Property to pull out common factors in the pairs of terms: 2[m(m + 8) – 3(m + 8)] = 0. Use the Distributive Property to pull out the common factor (m + 8). The quadratic is now completely factored: 2(m + 8)(m – 3) = 0. Apply the Zero Product Property: m + 8 = 0 or m – 3 = 0. Solve the individual equations: m = -8 or 3.   B) Incorrect. You probably either factored the quadratic incorrectly or you solved the individual equations incorrectly. The correct answer is m = -8 or 3.   C) Incorrect. You probably factored 2m2 + 10m as 2m(m + 5) and then set the factors equal to 0. However, the original equation is not equal to 0, it’s equal to 48. To use the Zero Product Property, one side must be 0. The correct answer is m = -8 or 3.   D) Incorrect. You probably factored 2m2 + 10m as 2m(m + 5) and then set the factors equal to 0. However, the original equation is not equal to 0, it’s equal to 48. To use the Zero Product Property, one side must be 0. The correct answer is m = -8 or 3.

 

 

 

We can find the solutions, or roots, of quadratic equations by setting one side equal to zero, factoring the polynomial, and then applying the Zero Product Property. The Zero Product Property states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0. When the product of factors equals zero, one or more of the factors must also equal zero. Once the polynomial is factored, set each factor equal to zero and solve them separately. The answers will be the set of possible solutions to the original equation.