5.4: 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
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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
Example A
Solve
Solution: The only difference between this problem and previous ones from the concepts before is the addition of the
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
This means that
Check your answer:
Example B
Solve
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.
Now, factor. The product of
Lastly, set each factor equal to zero and solve.
Check your work:
Example C
Solve
Solution: Here is an example of a quadratic equation without a constant term. The only thing we can do is take out the GCF.
Set the two factors equal to zero and solve.
Check:
Intro Problem Revisit When the ball hits the ground, the height h is 0. So the equation becomes
Let's factor and solve for t.
We need to find the factors of
Now set this factorization equal to zero and solve.
Because t represents the time, it must be positive. Only
Guided Practice
Solve the following equations by factoring.
1.
2.
3.
4.
Answers
1.
The factors are the same. When factoring a perfect square trinomial, the factors will always be the same. In this instance, the solutions for
When the two factors are the same, we call the solution for
2. Here, we need to get everything on the same side of the equals sign in order to factor.
Because there is no number in front of
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*}
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.
Practice
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?
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Image Attributions
Here you'll learn how to solve factorable quadratic equations for @$\begin{align*}x\end{align*}@$.