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Factorization of Quadratic Expressions with Negative Coefficients

Factor quadratics with positive and negative coefficients

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Factorization of Quadratic Expressions with Negative Coefficients

What if you had a quadratic expression like \begin{align*}x^2 - 3x - 10\end{align*}x23x10 or \begin{align*}-x^2 - 4x -4\end{align*}x24x4 in which some or all the coefficients were negative? How could you factor that expression? After completing this Concept, you'll be able to factor quadratic expressions like these for various negative coefficient values.

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CK-12 Foundation: 0909S Factoring Quadratic Expressions with Negative Coefficients

Guidance

In the previous concept, we saw how to factor quadratic expressions whose coefficients were all positive. In this concept we will now see what happens when we factor quadratic expressions where some of the coefficients are negative.

Factor when a = 1, b is Negative and c is Positive

Now let’s see how this method works if the middle coefficient is negative.

Example A

Factor \begin{align*}x^2 - 6x + 8\end{align*}x26x+8.

Solution

We are looking for an answer that is a product of two binomials in parentheses: \begin{align*}(x\;\;\;\;)(x\;\;\;\;)\end{align*}(x)(x)

When negative coefficients are involved, we have to remember that negative factors may be involved also. The number 8 can be written as the product of the following numbers:

\begin{align*}8 = 1 \cdot 8 \quad \quad \text{and} \quad \quad 1 + 8 = 9\end{align*}8=18and1+8=9

but also

\begin{align*}8 = (-1) \cdot (-8) \quad \quad \text{and} \quad \quad -1 + (-8) = -9\end{align*}8=(1)(8)and1+(8)=9

and

\begin{align*}8 = 2 \cdot 4 \quad \quad \text{and} \quad \quad 2 + 4 = 6\end{align*}8=24and2+4=6

but also

\begin{align*}8 = (-2) \cdot (-4) \quad \quad \text{and} \quad \quad -2 + (-4) = -6.\end{align*}8=(2)(4)and2+(4)=6.

The last option is the correct choice. The answer is \begin{align*}(x - 2)(x - 4)\end{align*}(x2)(x4). We can check to see if this is correct by multiplying \begin{align*}(x - 2)(x - 4)\end{align*}(x2)(x4):

\begin{align*}& \quad \quad \quad x - 2\\ & \underline{\;\;\;\;\;\;\;\;\;\;\;x - 4}\\ & \quad \ - \ 4x + 8\\ & \underline{x^2 - \ 2x\;\;\;\;\;\;\;}\\ & x^2 - \ 6x + 8\end{align*}x2x4  4x+8x2 2xx2 6x+8

The answer checks out.

Example B

Factor \begin{align*}x^2 - 17x + 16\end{align*}x217x+16.

Solution

We are looking for an answer that is a product of two binomials in parentheses: \begin{align*}(x\;\;\;\;)(x\;\;\;\;)\end{align*}(x)(x)

The number 16 can be written as the product of the following numbers:

\begin{align*}& 16 = 1 \cdot 16 && \text{and} && 1 + 16 = 17\\ & 16 = (-1) \cdot (-16) && \text{and} && -1 + (-16) = -17 \qquad (Correct \ choice)\\ & 16 = 2 \cdot 8 && \text{and} && 2 + 8 = 10\\ & 16 = (-2) \cdot (-8) && \text{and} && -2 + (-8) = -10\\ & 16 = 4 \cdot 4 && \text{and} && 4 + 4 = 8\\ & 16 = (-4) \cdot (-4) && \text{and} && -4 + (-4) = -8\end{align*}16=11616=(1)(16)16=2816=(2)(8)16=4416=(4)(4)andandandandandand1+16=171+(16)=17(Correct choice)2+8=102+(8)=104+4=84+(4)=8

The answer is \begin{align*}(x - 1)(x - 16)\end{align*}(x1)(x16).

In general, whenever \begin{align*}b\end{align*}b is negative and \begin{align*}a\end{align*}a and \begin{align*}c\end{align*}c are positive, the two binomial factors will have minus signs instead of plus signs.

Factor when a = 1 and c is Negative

Now let’s see how this method works if the constant term is negative.

Example C

Factor \begin{align*}x^2 + 2x - 15\end{align*}x2+2x15.

Solution

We are looking for an answer that is a product of two binomials in parentheses: \begin{align*}(x\;\;\;\;)(x\;\;\;\;\;)\end{align*}(x)(x)

Once again, 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 && \text{and} && -1 + 15 = 14\\ & -15 = 1 \cdot (-15) && \text{and} && 1 + (-15) = -14\\ & -15 = -3 \cdot 5 && \text{and} && -3 + 5 = 2 \qquad \qquad (Correct \ choice)\\ & -15 = 3 \cdot (-5) && \text{and} && 3 + (-5) = -2\end{align*}15=11515=1(15)15=3515=3(5)andandandand1+15=141+(15)=143+5=2(Correct choice)3+(5)=2

The answer is \begin{align*}(x - 3)(x +5)\end{align*}(x3)(x+5).

We can check to see if this is correct by multiplying:

\begin{align*}& \quad \quad \ \ x - \ 3\\ & \underline{\;\;\;\;\;\;\;\;\; x + \;5\;}\\ & \quad \quad 5x - 15\\ & \underline{x^2 - 3x\;\;\;\;\;\;\;\;}\\ & x^2 + 2x - 15\end{align*}  x 3x+55x15x23xx2+2x15

The answer checks out.

Example D

Factor \begin{align*}x^2 - 10x - 24\end{align*}x210x24.

Solution

We are looking for an answer that is a product of two binomials in parentheses: \begin{align*}(x\;\;\;\;)(x\;\;\;\;)\end{align*}(x)(x)

The number -24 can be written as the product of the following numbers:

\begin{align*}& -24 = -1 \cdot 24 && \text{and} && -1 + 24 = 23\\ & -24 = 1 \cdot (-24) && \text{and} && 1 + (-24) = -23\\ & -24 = -2 \cdot 12 && \text{and} && -2 + 12 = 10\\ & -24 = 2 \cdot (-12) && \text{and} && 2 + (-12) = -10 \qquad (Correct \ choice)\\ & -24 = -3 \cdot 8 && \text{and} && -3 + 8 = 5\\ & -24 = 3 \cdot (-8) && \text{and} && 3 + (-8) = -5\\ & -24 = -4 \cdot 6 && \text{and} && -4 + 6 = 2\\ & -24 = 4 \cdot (-6) && \text{and} && 4 + (-6) = -2\end{align*}24=12424=1(24)24=21224=2(12)24=3824=3(8)24=4624=4(6)andandandandandandandand1+24=231+(24)=232+12=102+(12)=10(Correct choice)3+8=53+(8)=54+6=24+(6)=2

The answer is \begin{align*}(x - 12) (x + 2)\end{align*}(x12)(x+2).

Factor when a = - 1

When \begin{align*}a = -1\end{align*}a=1, the best strategy is to factor the common factor of -1 from all the terms in the quadratic polynomial and then apply the methods you learned so far in this section

Example E

Factor \begin{align*}-x^2 + x + 6\end{align*}x2+x+6.

Solution

First factor the common factor of -1 from each term in the trinomial. Factoring -1 just changes the signs of each term in the expression:

\begin{align*}-x^2 + x + 6 = -(x^2 - x - 6)\end{align*}x2+x+6=(x2x6)

We’re looking for a product of two binomials in parentheses: \begin{align*}-(x\;\;\;\;)(x\;\;\;\;)\end{align*}(x)(x)

Now our job is to factor \begin{align*}x^2 - x - 6\end{align*}x2x6.

The number -6 can be written as the product of the following numbers:

\begin{align*}& -6 = -1 \cdot 6 && \text{and} && -1 + 6 = 5\\ & -6 = 1 \cdot (-6) && \text{and} && 1 + (-6) = -5\\ & -6 = -2 \cdot 3 && \text{and} && -2 + 3 = 1\\ & -6 = 2 \cdot (-3) && \text{and} && 2 + (-3) = -1 \qquad (Correct \ choice)\end{align*}6=166=1(6)6=236=2(3)andandandand1+6=51+(6)=52+3=12+(3)=1(Correct choice)

The answer is \begin{align*}-(x - 3)(x + 2)\end{align*}.

Watch this video for help with the Examples above.

CK-12 Foundation: Factoring Quadratic Expressions with Negative Coefficients

Guided Practice

Factor \begin{align*}x^2 + 34x - 35\end{align*}.

Solution

We are looking for an answer that is a product of two binomials in parentheses: \begin{align*}(x\;\;\;\;)(x\;\;\;\;)\end{align*}

The number -35 can be written as the product of the following numbers:

\begin{align*}& -35 = -1 \cdot 35 && \text{and} && -1 + 35 = 34 \qquad (Correct \ choice)\\ & -35 = 1 \cdot (-35) && \text{and} && 1 + (-35) = -34\\ & -35 = -5 \cdot 7 && \text{and} && -5 + 7 = 2\\ & -35 = 5 \cdot (-7) && \text{and} && 5 + (-7) = -2\end{align*}

The answer is \begin{align*}(x - 1)(x + 35)\end{align*}.

Explore More

Factor the following quadratic polynomials.

  1. \begin{align*}x^2 - 11x + 24\end{align*}
  2. \begin{align*}x^2 - 13x + 42\end{align*}
  3. \begin{align*}x^2 - 14x + 33\end{align*}
  4. \begin{align*}x^2 - 9x + 20\end{align*}
  5. \begin{align*}x^2 + 5x - 14\end{align*}
  6. \begin{align*}x^2 + 6x - 27\end{align*}
  7. \begin{align*}x^2 + 7x - 78\end{align*}
  8. \begin{align*}x^2 + 4x - 32\end{align*}
  9. \begin{align*}x^2 - 12x - 45\end{align*}
  10. \begin{align*}x^2 - 5x - 50\end{align*}
  11. \begin{align*}x^2 - 3x - 40\end{align*}
  12. \begin{align*}x^2 - x - 56\end{align*}
  13. \begin{align*}-x^2 - 2x - 1\end{align*}
  14. \begin{align*}-x^2 - 5x + 24\end{align*}
  15. \begin{align*}-x^2 + 18x - 72\end{align*}
  16. \begin{align*}-x^2 + 25x - 150\end{align*}
  17. \begin{align*}x^2 + 21x + 108\end{align*}
  18. \begin{align*}-x^2 + 11x - 30\end{align*}
  19. \begin{align*}x^2 + 12x - 64\end{align*}
  20. \begin{align*}x^2 - 17x - 60\end{align*}
  21. \begin{align*}x^2 + 5x - 36\end{align*}

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.9. 

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Vocabulary

Quadratic Polynomials

A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an x^2 term.

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