Question
5.54. Factor the following: (a) \( 2 x^{2}+3 x-2 \)
Ask by Martinez Bartlett. in the United States
Mar 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The factored form is \((x+2)(2x-1)\).
Solution
Factor the expression \( 2x^2+3x-2 \).
Factor the expression by following steps:
- step0: Factor:
\(2x^{2}+3x-2\)
- step1: Rewrite the expression:
\(2x^{2}+\left(-1+4\right)x-2\)
- step2: Calculate:
\(2x^{2}-x+4x-2\)
- step3: Rewrite the expression:
\(x\times 2x-x+2\times 2x-2\)
- step4: Factor the expression:
\(x\left(2x-1\right)+2\left(2x-1\right)\)
- step5: Factor the expression:
\(\left(x+2\right)\left(2x-1\right)\)
The factored form of the expression \(2x^{2}+3x-2\) is \((x+2)(2x-1)\).
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Simplify this solution The Deep Dive
To factor the quadratic \( 2x^{2} + 3x - 2 \), we look for two numbers that multiply to \( 2 \times -2 = -4 \) and add up to \( 3 \). The numbers \( 4 \) and \( -1 \) fit perfectly! We can rewrite the middle term: \[ 2x^2 + 4x - x - 2 \] Now, group the terms: \[ (2x^2 + 4x) + (-x - 2) \] Factor out common terms: \[ 2x(x + 2) - 1(x + 2) \] Now, we can see a common factor of \( (x + 2) \): \[ (2x - 1)(x + 2) \] So, \( 2x^{2} + 3x - 2 = (2x - 1)(x + 2) \). To verify, you can easily expand it back, and you'll get your original expression! Happy factoring!
