Question
CLHS Algebra 2 24.25 Unit 7 Test A Show all work for full credit. Place your final 2. \( \operatorname{simp} \) 1. Factor completely. \[ 2 x^{2}+13 x-7 \]
Ask by Logan Estrada. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 2x^{2} + 13x - 7 = (2x - 1)(x + 7) \)
Solution
We need to factor the quadratic expression:
2x² + 13x – 7
Step 1. Multiply the leading coefficient and the constant term:
a × c = 2 × (–7) = –14
Step 2. Find two numbers that multiply to –14 and add to the middle coefficient, 13.
These numbers are 14 and –1 because 14 × (–1) = –14 and 14 + (–1) = 13.
Step 3. Rewrite the middle term, 13x, using 14 and –1:
2x² + 14x – 1x – 7
Step 4. Factor by grouping:
Group the terms: (2x² + 14x) + (–1x – 7)
Factor out the greatest common factors (GCF) from each group:
• From 2x² + 14x, factor out 2x: 2x(x + 7)
• From –1x – 7, factor out –1: –1(x + 7)
Step 5. Notice the common factor (x + 7):
(2x – 1)(x + 7)
Thus, the completely factored form of 2x² + 13x – 7 is:
(2x – 1)(x + 7)
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Simplify this solution The Deep Dive
To factor the quadratic expression \(2x^2 + 13x - 7\), we look for two numbers that multiply to \(2 \times -7 = -14\) and add up to \(13\). The numbers \(14\) and \(-1\) fit this requirement. We can rewrite the middle term \(13x\) using these numbers: \[ 2x^2 + 14x - 1x - 7 \] Now group the terms: \[ (2x^2 + 14x) + (-1x - 7) \] Factor out the common factors: \[ 2x(x + 7) - 1(x + 7) \] Now, we can factor out \((x + 7)\): \[ (x + 7)(2x - 1) \] So, the complete factorization of \(2x^2 + 13x - 7\) is \((x + 7)(2x - 1)\).
