5.3: Factoring Special Quadratics
The total time, in hours, it takes a rower to paddle upstream, turn around and come back to her starting point is
Watch This
First, watch this video.
Khan Academy: U09_L2_TI_we1 Factoring Special Products 1
Then, watch the first part of this video, until about 3:10
James Sousa: Factoring a Difference of Squares
Guidance
There are a couple of special quadratics that, when factored, have a pattern.
Investigation: Multiplying the Square of a Binomal
1. Rewrite
2. FOIL your answer from Step 1. This is a perfect square trinomial.
3.
4. Apply the formula above to factoring
5. Now, plug
Investigation: Multiplying (a + b)(a - b)
1. FOIL
2. This is a difference of squares. The difference of squares will always factor to be
3. Apply the formula above to factoring
4. Now, plug
Example A
Factor
Solution: Using the formula from the investigation above, we need to first find the values of
Now, plugging
Alternate Method
Rewrite
Using the method from the previous two concepts, what are the two factors of -81 that add up to 0? 9 and -9
Therefore, the factors are
Example B
Factor
Solution: First, check for a GCF.
Now, double-check that the quadratic equation above fits into the perfect square trinomial formula.
Using
Alternate Method
First, find the GCF.
Then, find
\begin{align*}& 4(9x^2+{\color{blue}30x}+25)\\ & 4(9x^2+{\color{blue}15x+15x}+25)\\ & 4 \left[(9x^2+15x)+(15x+25) \right]\\ & 4 \left[3x(3x+5)+5(3x+5) \right]\\ & 4(3x+5)(3x+5) \ or \ 4(3x^2+5)\end{align*}
Again, notice that if you do not use the formula discovered in this concept, you can still factor and get the correct answer.
Example C
Factor \begin{align*}48x^2-147\end{align*}.
Solution: At first glance, this does not look like a difference of squares. 48 nor 147 are square numbers. But, if we take a 3 out of both, we have \begin{align*}3(16x^2-49)\end{align*}. 16 and 49 are both square numbers, so now we can use the formula.
\begin{align*}16x^2 &=a^2 \qquad 49=b^2\\ 4x &=a \qquad \quad 7=b\end{align*}
The factors are \begin{align*}3(4x-7)(4x+7)\end{align*}.
Intro Problem Revisit \begin{align*}18x^2 = 32\end{align*} can be rewritten as \begin{align*}18x^2-32 = 0\end{align*}, so factor \begin{align*}18x^2-32\end{align*}.
First, we must take greatest common factor of 2 out of both. We then have \begin{align*}2(9x^2-16)\end{align*}. 9 and 16 are both square numbers, so now we can use the formula.
\begin{align*}9x^2 &=a^2 \qquad 16=b^2\\ 3x &=a \qquad \quad 4=b\end{align*}
The factors are \begin{align*}2(3x-4)(3x+4)\end{align*}.
Finally, to find the time, set these factors equal to zero and solve \begin{align*}2(3x-4)(3x+4) = 0\end{align*}.
Because x represents the time, it must be positive. Only \begin{align*}(3x-4) = 0\end{align*} results in a positive value of x.
\begin{align*}x = \frac{4}{3} = 1.3333\end{align*} Therefore the round trip takes 1.3333 hours.
You will do more problems like this one in the next lesson.
Guided Practice
Factor the following quadratic equations.
1. \begin{align*}x^2-4\end{align*}
2. \begin{align*}2x^2-20x+50\end{align*}
3. \begin{align*}81x^2+144+64\end{align*}
Answers
1. \begin{align*}a = x\end{align*} and \begin{align*}b = 2\end{align*}. Therefore, \begin{align*}x^2-4=(x-2)(x+2)\end{align*}.
2. Factor out the GCF, 2. \begin{align*}2(x^2-10x+25)\end{align*}. This is now a perfect square trinomial with \begin{align*}a = x\end{align*} and \begin{align*}b = 5\end{align*}.
\begin{align*}2(x^2-10x+25)=2(x-5)^2.\end{align*}
3. This is a perfect square trinomial and no common factors. Solve for \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
\begin{align*}81x^2 &=a^2 \qquad 64=b^2\\ 9x &=a \qquad \quad 8=b\end{align*}
The factors are \begin{align*}(9x + 8)^2\end{align*}.
Vocabulary
- Perfect Square Trinomial
- A quadratic equation in the form \begin{align*}a^2+2ab+b^2\end{align*} or \begin{align*}a^2-2ab+b^2\end{align*}.
- Difference of Squares
- A quadratic equation in the form \begin{align*}a^2-b^2\end{align*}.
- Square Root
- A number, that when multiplied by itself produces another number. 3 is the square root of 9.
- Perfect Square
- A number that has a square root that is an integer. 25 is a perfect square.
Practice
- List the perfect squares that are less than 200.
- Why do you think there is no sum of squares formula?
Factor the following quadratics, if possible.
- \begin{align*}x^2-1\end{align*}
- \begin{align*}x^2+4x+4\end{align*}
- \begin{align*}16x^2-24x+9\end{align*}
- \begin{align*}-3x^2+36x-108\end{align*}
- \begin{align*}144x^2-49\end{align*}
- \begin{align*}196x^2+140x+25\end{align*}
- \begin{align*}100x^2+1\end{align*}
- \begin{align*}162x^2+72x+8\end{align*}
- \begin{align*}225-x^2\end{align*}
- \begin{align*}121-132x+36x^2\end{align*}
- \begin{align*}5x^2+100x-500\end{align*}
- \begin{align*}256x^2-676\end{align*}
- Error Analysis Spencer is given the following problem: Multiply \begin{align*}(2x-5)^2\end{align*}. Here is his work:
\begin{align*}(2x-5)^2=(2x)^2-5^2=4x^2-25\end{align*}
His teacher tells him the answer is \begin{align*}4x^2-20x+25\end{align*}. What did Spencer do wrong? Describe his error and correct the problem.
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Difference of Squares
A difference of squares is a quadratic equation in the form .Perfect Square
A perfect square is a number whose square root is an integer.Perfect Square Trinomial
A perfect square trinomial is a quadratic expression of the form (which can be rewritten as ) or (which can be rewritten as ).Quadratic form
A polynomial in quadratic form looks like a trinomial or binomial and can be factored like a quadratic expression.Square Root
The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.Image Attributions
Here you'll learn to factor perfect square trinomials and the difference of squares.