9.8: Factorization of Quadratic Expressions
Suppose your height above sea level in feet when traveling over a hill can be represented by the expression \begin{align*}-x^2 + 16x + 63\end{align*}, where \begin{align*}x\end{align*} is the horizontal distance traveled. If you wanted to factor this expression, could you do it? What would be the steps that you would follow? After completing this Concept, you'll know what to do in order to factor an expression such as this one.
Guidance
In this Concept, we will learn how to factor quadratic polynomials for different values of \begin{align*}a,\ b\end{align*}, and \begin{align*}c\end{align*}. In the last Concept, we factored common monomials, so you already know how to factor quadratic polynomials where \begin{align*}c=0\end{align*}.
Factoring Quadratic Expressions in Standard Form Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is \begin{align*}ax^2+bx+c\end{align*}, where \begin{align*}a,\ b\end{align*}, and \begin{align*}c\end{align*} are real numbers.
Example A
Factor \begin{align*}x^2+5x+6\end{align*}.
Solution: We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x+ \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x + \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.
To fill in the blanks, we want two numbers \begin{align*}m\end{align*} and \begin{align*}n\end{align*} that multiply to 6 and add to 5. A good strategy is to list the possible ways we can multiply two numbers to give us 6 and then see which of these pairs of numbers add to 5. The number six can be written as the product of:
\begin{align*}6&=1 \times 6 \qquad and \qquad 1+6=7\\ 6&=2 \times 3 \qquad and \qquad 2+3=5\end{align*}
So the answer is \begin{align*}(x+2)(x+3)\end{align*}.
We can check to see if this is correct by multiplying \begin{align*}(x+2)(x+3)\end{align*}.
\begin{align*}x\end{align*} is multiplied by \begin{align*}x\end{align*} and \begin{align*}3 =x^2+3x\end{align*}.
2 is multiplied by \begin{align*}x\end{align*} and \begin{align*}3 =2x+6\end{align*}.
Combine the like terms: \begin{align*}x^2+5x+6\end{align*}.
Example B
Factor \begin{align*}x^2-6x+8\end{align*}.
Solution: We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x+ \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x + \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.
The number 8 can be written as the product of the following numbers.
\begin{align*}8=1\cdot 8\end{align*} and \begin{align*}1+8=9\end{align*} Notice that these are two different choices.
\begin{align*}8&=(-1)(-8) && and && -1+(-8)=-9\\ 8&=2 \times 4 && and && 2+4=6\end{align*}
And
\begin{align*}8=(-2)\cdot (-4)\end{align*} and \begin{align*}-2+(-4)=-6 \leftarrow\end{align*} This is the correct choice.
The answer is \begin{align*}(x-2)(x-4)\end{align*}.
Example C
Factor \begin{align*}x^2+2x-15\end{align*}.
Solution: We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.
In this case, we must take the negative sign into account. The number –15 can be written as the product of the following numbers.
\begin{align*}-15=-1 \cdot 15\end{align*} and \begin{align*}-1+15=14\end{align*} Notice that these are two different choices.
And also,
\begin{align*}-15=1 \cdot (-15)\end{align*} and \begin{align*}1+(-15)=-14\end{align*} Notice that these are two different choices.
\begin{align*}-15=(-3) \times 5\end{align*} and \begin{align*}(-3)+5=2\end{align*} This is the correct choice.
\begin{align*}-15=3 \times (-5)\end{align*} and \begin{align*}3+(-5)=-2\end{align*}
The answer is \begin{align*}(x-3)(x+5)\end{align*}.
To Summarize:
A quadratic of the form \begin{align*}x^2+bx+c\end{align*} factors as a product of two binomials: \begin{align*}(x+m)(x+n)\end{align*}.
- If \begin{align*}b\end{align*} and \begin{align*}c\end{align*} are positive then both \begin{align*}m\end{align*} and \begin{align*}n\end{align*} 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*}.
Vocabulary
Quadratic polynomials: Quadratic polynomials are polynomials of degree 2.
Standard form of quadratic polynomials: The standard form of a quadratic polynomial is \begin{align*}ax^2+bx+c\end{align*}, where \begin{align*}a,\ b\end{align*}, and \begin{align*}c\end{align*} are real numbers.
Guided Practice
Factor \begin{align*} -x^2+x+6\end{align*}.
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.
\begin{align*}-x^2+x+6 = -(x^2-x-6)\end{align*}
We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.
Now our job is to factor \begin{align*}x^2-x-6\end{align*}.
The number –6 can be written as the product of the following numbers.
\begin{align*}&-6=(-1) \times 6 \qquad and \qquad (-1)+6=5\\ &-6=1 \times (-6) \qquad and \qquad 1+(-6)=-5\\ &-6=(-2) \times 3 \qquad and \qquad (-2)+3=1\\ &-6=2 \times (-3) \qquad and \qquad 2+(-3)=-1 \qquad This \ is \ the \ correct \ choice.\end{align*}
The answer is \begin{align*}-(x-3)(x+2)\end{align*}.
Practice
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*}.
Quadratic Polynomials
A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an term.standard form of quadratic polynomials
The standard form of a quadratic polynomial is , where , and are real numbers.Greatest Common Factor
The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.linear factors
Linear factors are expressions of the form where and are real numbers.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Quadratic Formula
The quadratic formula states that for any quadratic equation in the form , .Trinomial
A trinomial is a mathematical expression with three terms.Image Attributions
Here you'll learn to determine what the factors of a quadratic expression are.