Question
Factor the trinomial completely. \[ x^{2}+x-72 \]
Ask by Weaver Greene. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The trinomial \( x^{2} + x - 72 \) factors to \( (x - 8)(x + 9) \).
Solution
To factor the trinomial \( x^{2} + x - 72 \) completely, follow these steps:
1. **Identify the coefficients:**
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 1 \) (coefficient of \( x \))
- \( c = -72 \) (constant term)
2. **Find two numbers that multiply to \( a \times c = 1 \times (-72) = -72 \) and add up to \( b = 1 \).**
Let's list the factor pairs of \(-72\):
- \( 1 \) and \( -72 \) (sum: \(-71\))
- \( -1 \) and \( 72 \) (sum: \(71\))
- \( 2 \) and \( -36 \) (sum: \(-34\))
- \( -2 \) and \( 36 \) (sum: \(34\))
- \( 3 \) and \( -24 \) (sum: \(-21\))
- \( -3 \) and \( 24 \) (sum: \(21\))
- \( 4 \) and \( -18 \) (sum: \(-14\))
- \( -4 \) and \( 18 \) (sum: \(14\))
- \( 6 \) and \( -12 \) (sum: \(-6\))
- \( -6 \) and \( 12 \) (sum: \(6\))
- \( 8 \) and \( -9 \) (sum: \(-1\))
- \( -8 \) and \( 9 \) (sum: \(1\))
The pair that adds up to \(1\) is \(-8\) and \(9\).
3. **Rewrite the middle term using these numbers:**
\[
x^{2} + 9x - 8x - 72
\]
4. **Factor by grouping:**
\[
x(x + 9) - 8(x + 9)
\]
\[
(x - 8)(x + 9)
\]
**Final Factored Form:**
\[
(x - 8)(x + 9)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To factor the trinomial \( x^{2}+x-72 \), we look for two numbers that multiply to -72 (the constant term) and add to 1 (the coefficient of the linear term). The numbers 9 and -8 fit the bill because \( 9 \times -8 = -72 \) and \( 9 + (-8) = 1 \). Therefore, we can factor the trinomial as: \[ (x + 9)(x - 8) \] To factor the trinomial \( x^2 + x - 72 \) completely, we can double-check our factorization by expanding: \[ (x + 9)(x - 8) = x^2 - 8x + 9x - 72 = x^2 + x - 72 \] Thus, the final factorization is: \[ (x + 9)(x - 8) \]
