Question
20) \( 2 x^{2}-5 x-3 \)
Ask by Conner Potter. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \(2x^2 - 5x - 3\) factors to \((2x + 1)(x - 3)\).
Solution
1. We start with the quadratic expression:
\[
2x^2 - 5x - 3
\]
2. To factor this quadratic, we look for two numbers that multiply to \(a \times c = 2 \times (-3) = -6\) and add to \(b = -5\).
3. The two numbers that satisfy these conditions are \(-6\) and \(1\) since:
\[
-6 \times 1 = -6 \quad \text{and} \quad -6 + 1 = -5
\]
4. Rewrite the middle term, \(-5x\), using \(-6x\) and \(1x\):
\[
2x^2 - 6x + x - 3
\]
5. Factor by grouping:
First group:
\[
2x^2 - 6x = 2x(x - 3)
\]
Second group:
\[
x - 3 = 1(x - 3)
\]
6. Factor out the common factor \((x - 3)\):
\[
2x(x - 3) + 1(x - 3) = (2x + 1)(x - 3)
\]
7. Thus, the factored form of the expression is:
\[
(2x+1)(x-3)
\]
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Simplify this solution The Deep Dive
To factor the quadratic expression \(2x^2 - 5x - 3\), think of it like a treasure hunt for two numbers that multiply to \(-6\) (the product of \(2\) and \(-3\)) and add to \(-5\). After some exploration, you'll find that \(-6\) and \(1\) do the trick! Rewriting the expression gives us \(2x^2 - 6x + x - 3\). Now, group and factor: take out the common factors to reveal \((2x + 1)(x - 3)\). Happy factoring! As a little side note, you can always check your factorization by expanding it back out to make sure it's the same as the original equation. If you get what you started with, then you've hit the jackpot!
