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# Solving Problems by Factoring

## Factor and use the zero product rule

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Practice Solving Problems by Factoring
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The height of a ball that is thrown straight up in the air from a height of 2 meters above the ground with a velocity of 9 meters per second is given by the quadratic equation h=5t2+9t+2\begin{align*}h = -5t^2 + 9t + 2\end{align*}, where t is the time in seconds. How long does it take the ball to hit the ground?

### Watch This

Watch the first part of this video, until about 4:40.

### Guidance

In this lesson we have not actually solved for x\begin{align*}x\end{align*}. Now, we will apply factoring to solving a quadratic equation. It adds one additional step to the end of what you have already been doing. Let’s go through an example.

#### Example A

Solve x29x+18=0\begin{align*}x^2-9x+18=0\end{align*} by factoring.

Solution: The only difference between this problem and previous ones from the concepts before is the addition of the =\begin{align*}=\end{align*} sign. Now that this is present, we need to solve for x\begin{align*}x\end{align*}. We can still factor the way we always have. Because a=1\begin{align*}a = 1\end{align*}, determine the two factors of 18 that add up to -9.

x29x+18(x6)(x3)=0=0

Now, we have two factors that, when multiplied, equal zero. Recall that when two numbers are multiplied together and one of them is zero, the product is always zero.

Zero-Product Property: If ab=0\begin{align*}ab = 0\end{align*}, then a=0\begin{align*}a = 0\end{align*} or b=0\begin{align*}b = 0\end{align*}.

This means that x6=0\begin{align*}x-6 = 0\end{align*} OR x3=0\begin{align*}x-3 = 0\end{align*}. Therefore, x=6\begin{align*}x = 6\end{align*} or x=3\begin{align*} x = 3\end{align*}. There will always be the same number of solutions as factors.

629(6)+183654+18=0or329(3)+18=0=0927+18=0

#### Example B

Solve 6x2+x4=11\begin{align*}6x^2+x-4=11\end{align*} by factoring.

Solution: At first glance, this might not look factorable to you. However, before we factor, we must combine like terms. Also, the Zero-Product Property tells us that in order to solve for the factors, one side of the equation must be zero.

6x2+x4=1111=116x2+x15=0

Now, factor. The product of ac\begin{align*}ac\end{align*} is -90. What are the two factors of -90 that add up to 1? 10 and -9. Expand the x\begin{align*}x-\end{align*}term and factor.

6x2+x156x29x+10x153x(2x3)+5(2x3)(2x3)(3x+5)=0=0=0=0

Lastly, set each factor equal to zero and solve.

2x32xx=0 3x+5=0=3or3x=5=32x=53

6(32)2+324694+324272+324154=11  6(53)2534=11=11or  6259534=11=11503534=11=11154=11

#### Example C

Solve 10x225x=0\begin{align*}10x^2-25x=0\end{align*} by factoring.

Solution: Here is an example of a quadratic equation without a constant term. The only thing we can do is take out the GCF.

10x225x5x(2x5)=0=0

Set the two factors equal to zero and solve.

5xx=02x5=0=0or  2x=5x=52

Check:

10(0)225(0)=010(52)225(52)=0  0=0or102541252=012521252=0

Intro Problem Revisit When the ball hits the ground, the height h is 0. So the equation becomes 0=5t2+9t+2\begin{align*}0 = -5t^2 + 9t + 2\end{align*}.

Let's factor and solve for t. 5t2+9t+2\begin{align*}-5t^2 + 9t + 2\end{align*}

We need to find the factors of 10\begin{align*}-10\end{align*} that add up to 9. Testing the possibilities, we find 10 and -1 to be the correct combination.

5t2+10tt+2\begin{align*}-5t^2 + 10t - t + 2\end{align*} = (5t2+10t)+(t+2)\begin{align*}(-5t^2 + 10t)+ (-t + 2)\end{align*} = 5t(t+2)+(t+2)\begin{align*}5t(-t + 2)+ (-t + 2)\end{align*} = (5t+1)(t+2)\begin{align*}(5t + 1)(-t + 2)\end{align*}

Now set this factorization equal to zero and solve.

(5t+1)(t+2)=0\begin{align*}(5t + 1)(-t + 2)=0\end{align*}

Because t represents the time, it must be positive. Only (t+2)=0\begin{align*}(-t + 2)=0\end{align*} results in a positive value.

t=2\begin{align*}t = 2\end{align*}, therefore it takes the ball 2 seconds to reach the ground.

### Guided Practice

Solve the following equations by factoring.

1. 4x212x+9=0\begin{align*}4x^2-12x+9=0\end{align*}

2. x25x=6\begin{align*}x^2-5x=6\end{align*}

3. 8x20x2=0\begin{align*}8x-20x^2=0\end{align*}

4. 12x2+13x+7=124x\begin{align*}12x^2+13x+7=12-4x\end{align*}

1. ac=36\begin{align*}ac = 36\end{align*}. The factors of 36 that also add up to -12 are -6 and -6. Expand the x\begin{align*}x-\end{align*}term and factor.

4x212x+94x26x6x+92x(2x3)3(2x3)(2x3)(2x3)=0=0=0=0

The factors are the same. When factoring a perfect square trinomial, the factors will always be the same. In this instance, the solutions for x\begin{align*}x\end{align*} will also be the same. Solve for x\begin{align*}x\end{align*}.

2x32xx=0=3=32

When the two factors are the same, we call the solution for x\begin{align*}x\end{align*} a double root because it is the solution twice.

2. Here, we need to get everything on the same side of the equals sign in order to factor.

x25xx25x6=6=0

Because there is no number in front of x2\begin{align*}x^2\end{align*}, we need to find the factors of -6 that add up to -5.

(x6)(x+1)=0

Solving each factor for \begin{align*}x\end{align*}, we get that \begin{align*}x = 6\end{align*} or \begin{align*}x = -1\end{align*}.

3. Here there is no constant term. Find the GCF to factor.

Solve each factor for \begin{align*}x\end{align*}.

4. This problem is slightly more complicated than #2. Combine all like terms onto the same side of the equals sign so that one side is zero.

\begin{align*}ac = -60\end{align*}. The factors of -60 that add up to 17 are 20 and -3. Expand the \begin{align*}x-\end{align*}term and factor.

Solve each factor for \begin{align*}x\end{align*}.

### Explore More

Solve the following quadratic equations by factoring, if possible.

1. \begin{align*}x^2+8x-9=0\end{align*}
2. \begin{align*}x^2+6x=0\end{align*}
3. \begin{align*}2x^2-5x=12\end{align*}
4. \begin{align*}12x^2+7x-10=0\end{align*}
5. \begin{align*}x^2=9\end{align*}
6. \begin{align*}30x+25=-9x^2\end{align*}
7. \begin{align*}2x^2+x-5=0\end{align*}
8. \begin{align*}16x=32x^2\end{align*}
9. \begin{align*}3x^2+28x=-32\end{align*}
10. \begin{align*}36x^2-48=1\end{align*}
11. \begin{align*}6x^2+x=4\end{align*}
12. \begin{align*}5x^2+12x+4=0\end{align*}

Challenge Solve these quadratic equations by factoring. They are all factorable.

1. \begin{align*}8x^2+8x-5=10-6x\end{align*}
2. \begin{align*}-18x^2=48x+14\end{align*}
3. \begin{align*}36x^2-24=96x-39\end{align*}
4. Real Life Application George is helping his dad build a fence for the backyard. The total area of their backyard is 1600 square feet. The width of the house is half the length of the yard, plus 7 feet. How much fencing does George’s dad need to buy?

### Vocabulary Language: English

Double Root

Double Root

A solution that is repeated twice.
solution

solution

A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.