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 \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?
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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 @$\begin{align*}x\end{align*}@$. 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 @$\begin{align*}x^2-9x+18=0\end{align*}@$ 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 @$\begin{align*}x\end{align*}@$. We can still factor the way we always have. Because @$\begin{align*}a = 1\end{align*}@$, determine the two factors of 18 that add up to -9.
@$$\begin{align*}x^2-9x+18 &= 0\\ (x-6)(x-3) &= 0\end{align*}@$$
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 @$\begin{align*}ab = 0\end{align*}@$, then @$\begin{align*}a = 0\end{align*}@$ or @$\begin{align*}b = 0\end{align*}@$.
This means that @$\begin{align*}x-6 = 0\end{align*}@$ OR @$\begin{align*}x-3 = 0\end{align*}@$. Therefore, @$\begin{align*}x = 6\end{align*}@$ or @$\begin{align*} x = 3\end{align*}@$. There will always be the same number of solutions as factors.
Check your answer:
@$$\begin{align*} 6^2-9(6)+18 &=0 \quad or \quad 3^2-9(3)+18=0\\ 36-54+18 &=0 \qquad \quad \quad 9-27+18=0 \end{align*}@$$
Example B
Solve @$\begin{align*}6x^2+x-4=11\end{align*}@$ 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.
@$$\begin{align*}& \ 6x^2+x-4 = \bcancel{11}\\ & \underline{\; \; \; \; \; \; \; \; \; \; \; \; \; -11 = - \bcancel{11} \; \;}\\ & 6x^2+x-15=0\end{align*}@$$
Now, factor. The product of @$\begin{align*}ac\end{align*}@$ is -90. What are the two factors of -90 that add up to 1? 10 and -9. Expand the @$\begin{align*}x-\end{align*}@$term and factor.
@$$\begin{align*}6x^2+x-15 &= 0\\ 6x^2-9x+10x-15 &= 0\\ 3x(2x-3)+5(2x-3) &= 0\\ (2x-3)(3x+5) &= 0\end{align*}@$$
Lastly, set each factor equal to zero and solve.
@$$\begin{align*}2x-3 &=0 \qquad \ 3x+5 = 0\\ 2x &=3 \quad or \quad \quad 3x=-5\\ x &=\frac{3}{2} \qquad \qquad x=-\frac{5}{3}\end{align*}@$$
Check your work:
@$$\begin{align*}6 \left(\frac{3}{2}\right)^2 +\frac{3}{2}-4 &= 11 \qquad \ \ 6 \left(- \frac{5}{3}\right)^2 -\frac{5}{3}-4 = 11 \\ 6 \cdot \frac{9}{4}+\frac{3}{2}-4 &=11 \quad or \quad \ \ 6 \cdot \frac{25}{9}-\frac{5}{3}-4 =11 \\ \frac{27}{2}+\frac{3}{2}-4 &=11 \qquad \qquad \quad \frac{50}{3}-\frac{5}{3}-4=11\\ 15-4 &=11 \qquad \qquad \qquad \quad 15-4=11\end{align*}@$$
Example C
Solve @$\begin{align*}10x^2-25x=0\end{align*}@$ 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.
@$$\begin{align*}10x^2-25x &= 0\\ 5x(2x-5) &= 0\end{align*}@$$
Set the two factors equal to zero and solve.
@$$\begin{align*}5x &=0 \qquad 2x-5=0\\ x &=0 \quad or \quad \ \ 2x=5\\ & \qquad \qquad \qquad x=\frac{5}{2}\end{align*}@$$
Check:
@$$\begin{align*}& 10(0)^2-25(0) = 0 \qquad \qquad 10\left(\frac{5}{2}\right)^2- 25 \left(\frac{5}{2}\right)=0\\ & \qquad \qquad \quad \ \ 0 = 0 \qquad or \quad \quad 10 \cdot \frac{25}{4}-\frac{125}{2}=0 \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \frac{125}{2} - \frac{125}{2} =0\end{align*}@$$
Intro Problem Revisit When the ball hits the ground, the height h is 0. So the equation becomes @$\begin{align*}0 = -5t^2 + 9t + 2\end{align*}@$.
Let's factor and solve for t. @$\begin{align*}-5t^2 + 9t + 2\end{align*}@$
We need to find the factors of @$\begin{align*}-10\end{align*}@$ that add up to 9. Testing the possibilities, we find 10 and -1 to be the correct combination.
@$\begin{align*}-5t^2 + 10t - t + 2\end{align*}@$ = @$\begin{align*}(-5t^2 + 10t)+ (-t + 2)\end{align*}@$ = @$\begin{align*}5t(-t + 2)+ (-t + 2)\end{align*}@$ = @$\begin{align*}(5t + 1)(-t + 2)\end{align*}@$
Now set this factorization equal to zero and solve.
@$\begin{align*}(5t + 1)(-t + 2)=0\end{align*}@$
Because t represents the time, it must be positive. Only @$\begin{align*}(-t + 2)=0\end{align*}@$ results in a positive value.
@$\begin{align*}t = 2\end{align*}@$, therefore it takes the ball 2 seconds to reach the ground.
Guided Practice
Solve the following equations by factoring.
1. @$\begin{align*}4x^2-12x+9=0\end{align*}@$
2. @$\begin{align*}x^2-5x=6\end{align*}@$
3. @$\begin{align*}8x-20x^2=0\end{align*}@$
4. @$\begin{align*}12x^2+13x+7=12-4x\end{align*}@$
Answers
1. @$\begin{align*}ac = 36\end{align*}@$. The factors of 36 that also add up to -12 are -6 and -6. Expand the @$\begin{align*}x-\end{align*}@$term and factor.
@$$\begin{align*}4x^2-12x+9 &= 0\\ 4x^2-6x-6x+9 &= 0\\ 2x(2x-3)-3(2x-3) &= 0\\ (2x-3)(2x-3) &=0\end{align*}@$$
The factors are the same. When factoring a perfect square trinomial, the factors will always be the same. In this instance, the solutions for @$\begin{align*}x\end{align*}@$ will also be the same. Solve for @$\begin{align*}x\end{align*}@$.
@$$\begin{align*}2x-3 &= 0\\ 2x &= 3\\ x &= \frac{3}{2}\end{align*}@$$
When the two factors are the same, we call the solution for @$\begin{align*}x\end{align*}@$ 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.
@$$\begin{align*}x^2-5x &= 6\\ x^2-5x-6 &= 0\end{align*}@$$
Because there is no number in front of @$\begin{align*}x^2\end{align*}@$, we need to find the factors of -6 that add up to -5.
@$$\begin{align*}(x-6)(x+1)=0\end{align*}@$$
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.
@$$\begin{align*}8x-20x^2 &= 0\\ 4x(2-5x) &= 0\end{align*}@$$
Solve each factor for @$\begin{align*}x\end{align*}@$.
@$$\begin{align*}4x &=0 \qquad 2-5x=0\\ x &=0 \quad or \qquad \ 2=5x\\ & \qquad \qquad \qquad \frac{2}{5}=x\end{align*}@$$
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.
@$$\begin{align*}12x^2+13x+7 &= 12-4x\\ 12x^2+17x-5 &= 0\end{align*}@$$
@$\begin{align*}ac = -60\end{align*}@$. The factors of -60 that add up to 17 are 20 and -3. Expand the @$\begin{align*}x-\end{align*}@$term and factor.
@$$\begin{align*}12x^2+17x-5 &= 0\\ 12 x^2+20x-3x-5 &= 0\\ 4x(3x+5)-1(3x+5) &=0\\ (3x+5)(4x-1) &= 0\end{align*}@$$
Solve each factor for @$\begin{align*}x\end{align*}@$.
@$$\begin{align*}3x+5 &=0 \qquad 4x-1 = 0 \\ 3x &= -5 \quad or \quad 4x=1\\ x &= -\frac{5}{3} \qquad \quad x = \frac{1}{4}\end{align*}@$$
Explore More
Solve the following quadratic equations by factoring, if possible.
- @$\begin{align*}x^2+8x-9=0\end{align*}@$
- @$\begin{align*}x^2+6x=0\end{align*}@$
- @$\begin{align*}2x^2-5x=12\end{align*}@$
- @$\begin{align*}12x^2+7x-10=0\end{align*}@$
- @$\begin{align*}x^2=9\end{align*}@$
- @$\begin{align*}30x+25=-9x^2\end{align*}@$
- @$\begin{align*}2x^2+x-5=0\end{align*}@$
- @$\begin{align*}16x=32x^2\end{align*}@$
- @$\begin{align*}3x^2+28x=-32\end{align*}@$
- @$\begin{align*}36x^2-48=1\end{align*}@$
- @$\begin{align*}6x^2+x=4\end{align*}@$
- @$\begin{align*}5x^2+12x+4=0\end{align*}@$
Challenge Solve these quadratic equations by factoring. They are all factorable.
- @$\begin{align*}8x^2+8x-5=10-6x\end{align*}@$
- @$\begin{align*}-18x^2=48x+14\end{align*}@$
- @$\begin{align*}36x^2-24=96x-39\end{align*}@$
- 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?