9.5: Factoring Quadratic Expressions
In this lesson, we will learn how to factor quadratic polynomials for different values of
Factoring Quadratic Expressions in Standard From
Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is
Example 1: Factor
Solution: We are looking for an answer that is a product of two binomials in parentheses:
To fill in the blanks, we want two numbers
So the answer is
We can check to see if this is correct by multiplying
2 is multiplied by
Combine the like terms:
Example 2: Factor
Solution: We are looking for an answer that is a product of the two parentheses
The number 8 can be written as the product of the following numbers.
And
The answer is
Example 3: Factor
Solution: We are looking for an answer that is a product of two parentheses
In this case, we must take the negative sign into account. The number –15 can be written as the product of the following numbers.
And also,
The answer is
Example 4: Factor
Solution: First factor the common factor of –1 from each term in the trinomial. Factoring –1 changes the signs of each term in the expression.
We are looking for an answer that is a product of two parentheses
Now our job is to factor
The number –6 can be written as the product of the following numbers.
The answer is
To Summarize:
A quadratic of the form
- If
b andc are positive then bothm andn are positive.- Example \begin{align*}x^2+8x+12\end{align*} factors as \begin{align*}(x+6)(x+2)\end{align*}.
- If \begin{align*}b\end{align*} is negative and \begin{align*}c\end{align*} is positive then both \begin{align*}m\end{align*} and \begin{align*}n\end{align*} are negative.
- Example \begin{align*}x^2-6x+8\end{align*} factors as \begin{align*}(x-2)(x-4)\end{align*}.
- If \begin{align*}c\end{align*} is negative then either \begin{align*}m\end{align*} is positive and \begin{align*}n\end{align*} is negative or vice-versa.
- Example \begin{align*}x^2+2x-15\end{align*} factors as \begin{align*}(x+5)(x-3)\end{align*}.
- Example \begin{align*}x^2+34x-35\end{align*} factors as \begin{align*}(x+35)(x-1)\end{align*}.
- If \begin{align*}a=-1\end{align*}, factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form \begin{align*}-(x+m)(x+n)\end{align*}.
- Example \begin{align*}-x^2+x+6\end{align*} factors as \begin{align*}-(x-3)(x+2)\end{align*}.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Factoring Quadratic Equations (16:30)
Factor the following quadratic polynomials.
- \begin{align*}x^2+10x+9\end{align*}
- \begin{align*}x^2+15x+50\end{align*}
- \begin{align*}x^2+10x+21\end{align*}
- \begin{align*}x^2+16x+48\end{align*}
- \begin{align*}x^2-11x+24\end{align*}
- \begin{align*}x^2-13x+42\end{align*}
- \begin{align*}x^2-14x+33\end{align*}
- \begin{align*}x^2-9x+20\end{align*}
- \begin{align*}x^2+5x-14\end{align*}
- \begin{align*}x^2+6x-27\end{align*}
- \begin{align*}x^2+7x-78\end{align*}
- \begin{align*}x^2+4x-32\end{align*}
- \begin{align*}x^2-12x-45\end{align*}
- \begin{align*}x^2-5x-50\end{align*}
- \begin{align*}x^2-3x-40\end{align*}
- \begin{align*}x^2-x-56\end{align*}
- \begin{align*}-x^2-2x-1\end{align*}
- \begin{align*}-x^2-5x+24\end{align*}
- \begin{align*}-x^2+18x-72\end{align*}
- \begin{align*}-x^2+25x-150\end{align*}
- \begin{align*}x^2+21x+108\end{align*}
- \begin{align*}-x^2+11x-30\end{align*}
- \begin{align*}x^2+12x-64\end{align*}
- \begin{align*}x^2-17x-60\end{align*}
Mixed Review
- Evaluate \begin{align*}f(2)\end{align*} when \begin{align*}f(x)=\frac{1}{2} x^2-6x+4\end{align*}.
- The Nebraska Department of Roads collected the following data regarding mobile phone distractions in traffic crashes by teen drivers.
- Plot the data as a scatter plot.
- Fit a line to this data.
- Predict the number of teenage traffic accidents attributable to cell phones in the year 2012.
Year (\begin{align*}y\end{align*}) | Total (\begin{align*}n\end{align*}) |
---|---|
2002 | 41 |
2003 | 43 |
2004 | 47 |
2005 | 38 |
2006 | 36 |
2007 | 40 |
2008 | 42 |
2009 | 42 |
- Simplify \begin{align*}\sqrt{405}\end{align*}.
- Graph the following on a number line: \begin{align*}-\pi, \sqrt{2}, \frac{5}{3}, - \frac{3}{10}, \sqrt{16}\end{align*}.
- What is the multiplicative inverse of \begin{align*}\frac{9}{4}\end{align*}?
Quick Quiz
- Name the following polynomial. State its degree and leading coefficient \begin{align*}6x^2 y^4 z+6x^6-2y^5+11xyz^4\end{align*}.
- Simplify \begin{align*}(a^2 b^2 c+11abc^5 )+(4abc^5-3a^2 b^2 c+9abc)\end{align*}.
- A rectangular solid has dimensions \begin{align*}(a+2)\end{align*} by \begin{align*}(a+4)\end{align*} by \begin{align*}(3a)\end{align*}. Find its volume.
- Simplify \begin{align*}-3hjk^3 (h^2 j^4 k+6hk^2)\end{align*}.
- Find the solutions to \begin{align*}(x-3)(x+4)(2x-1)=0\end{align*}.
- Multiply \begin{align*}(a-9b)(a+9b)\end{align*}.
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