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Solving Problems by Factoring

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Solving Quadratics using Factoring

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 = -5t^2 + 9t + 2 , 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.

Khan Academy: Solving Quadratic Equations by Factoring.avi

Guidance

In this lesson we have not actually solved for x . 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 x^2-9x+18=0 by factoring.

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

x^2-9x+18 &= 0\\(x-6)(x-3) &= 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 , then a = 0 or b = 0 .

This means that x-6 = 0 OR x-3 = 0 . Therefore, x = 6 or  x = 3 . There will always be the same number of solutions as factors.

Check your answer:

 6^2-9(6)+18 &=0 \quad or \quad 3^2-9(3)+18=0\\36-54+18 &=0 \qquad \quad \quad 9-27+18=0

Example B

Solve 6x^2+x-4=11 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.

& \ 6x^2+x-4 = \bcancel{11}\\& \underline{\; \; \; \; \; \; \; \; \; \; \; \; \; -11 = - \bcancel{11} \; \;}\\& 6x^2+x-15=0

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

6x^2+x-15 &= 0\\6x^2-9x+10x-15 &= 0\\3x(2x-3)+5(2x-3) &= 0\\(2x-3)(3x+5) &= 0

Lastly, set each factor equal to zero and solve.

2x-3 &=0 \qquad \ 3x+5 = 0\\2x &=3 \quad or \quad \quad 3x=-5\\x &=\frac{3}{2} \qquad \qquad x=-\frac{5}{3}

Check your work:

6 \left(\frac{3}{2}\right)^2 +\frac{3}{2}-4 &= 11 \qquad \ \ 6 \left(- \frac{5}{3}\right)^2 -\frac{5}{3}-4 = 11 \\6 \cdot \frac{9}{4}+\frac{3}{2}-4 &=11 \quad or \quad \ \ 6 \cdot \frac{25}{9}-\frac{5}{3}-4 =11 \\\frac{27}{2}+\frac{3}{2}-4 &=11 \qquad \qquad \quad \frac{50}{3}-\frac{5}{3}-4=11\\15-4 &=11 \qquad \qquad \qquad \quad 15-4=11

Example C

Solve 10x^2-25x=0 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.

10x^2-25x &= 0\\5x(2x-5) &= 0

Set the two factors equal to zero and solve.

5x &=0 \qquad 2x-5=0\\x &=0 \quad or \quad \ \ 2x=5\\& \qquad \qquad \qquad x=\frac{5}{2}

Check:

& 10(0)^2-25(0) = 0 \qquad \qquad 10\left(\frac{5}{2}\right)^2- 25 \left(\frac{5}{2}\right)=0\\& \qquad \qquad \quad \ \ 0 = 0 \qquad or \quad \quad 10 \cdot \frac{25}{4}-\frac{125}{2}=0 \\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \frac{125}{2} - \frac{125}{2} =0

Intro Problem Revisit When the ball hits the ground, the height h is 0. So the equation becomes 0 = -5t^2 + 9t + 2 .

Let's factor and solve for t . -5t^2 + 9t + 2

We need to find the factors of -10 that add up to 9. Testing the possibilities, we find 10 and -1 to be the correct combination.

-5t^2 + 10t - t + 2 = (-5t^2 + 10t)+ (-t + 2) = 5t(-t + 2)+ (-t + 2) = (5t + 1)(-t + 2)

Now set this factorization equal to zero and solve.

(5t + 1)(-t + 2)=0

Because t represents the time, it must be positive. Only (-t + 2)=0 results in a positive value.

t = 2 , therefore it takes the ball 2 seconds to reach the ground.

Guided Practice

Solve the following equations by factoring.

1. 4x^2-12x+9=0

2. x^2-5x=6

3. 8x-20x^2=0

4. 12x^2+13x+7=12-4x

Answers

1. ac = 36 . The factors of 36 that also add up to -12 are -6 and -6. Expand the x- term and factor.

4x^2-12x+9 &= 0\\4x^2-6x-6x+9 &= 0\\2x(2x-3)-3(2x-3) &= 0\\(2x-3)(2x-3) &=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 will also be the same. Solve for x .

2x-3 &= 0\\2x &= 3\\x &= \frac{3}{2}

When the two factors are the same, we call the solution for x 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.

x^2-5x &= 6\\x^2-5x-6 &= 0

Because there is no number in front of x^2 , we need to find the factors of -6 that add up to -5.

(x-6)(x+1)=0

Solving each factor for x , we get that x = 6 or x = -1 .

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

8x-20x^2 &= 0\\4x(2-5x) &= 0

Solve each factor for x .

4x &=0 \qquad 2-5x=0\\x &=0 \quad or \qquad \ 2=5x\\& \qquad \qquad \qquad \frac{2}{5}=x

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.

12x^2+13x+7 &= 12-4x\\12x^2+17x-5 &= 0

ac = -60 . The factors of -60 that add up to 17 are 20 and -3. Expand the x- term and factor.

12x^2+17x-5 &= 0\\12 x^2+20x-3x-5 &= 0\\4x(3x+5)-1(3x+5) &=0\\(3x+5)(4x-1) &= 0

Solve each factor for x .

3x+5 &=0 \qquad 4x-1 = 0 \\3x &= -5 \quad or \quad 4x=1\\x &= -\frac{5}{3} \qquad \quad x = \frac{1}{4}

Vocabulary

Solution
The answer to an equation. With quadratic equations, solutions can also be called zeros or roots .
Double Root
A solution that is repeated twice.

Explore More

Solve the following quadratic equations by factoring, if possible.

  1. x^2+8x-9=0
  2. x^2+6x=0
  3. 2x^2-5x=12
  4. 12x^2+7x-10=0
  5. x^2=9
  6. 30x+25=-9x^2
  7. 2x^2+x-5=0
  8. 16x=32x^2
  9. 3x^2+28x=-32
  10. 36x^2-48=1
  11. 6x^2+x=4
  12. 5x^2+12x+4=0

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

  1. 8x^2+8x-5=10-6x
  2. -18x^2=48x+14
  3. 36x^2-24=96x-39
  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?

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