Asked by jack
Factor 8x^2−3x−15
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Answered by
GPT-4o mini
To factor the quadratic expression \(8x^2 - 3x - 15\), we can use the method of factoring by grouping or search for two numbers that multiply to \(a \cdot c\) (where \(a\) is the coefficient of \(x^2\) and \(c\) is the constant term) and add to \(b\) (the coefficient of \(x\)).
Here, \(a = 8\), \(b = -3\), and \(c = -15\).
1. First, calculate \(a \cdot c = 8 \cdot (-15) = -120\).
2. Next, we need to find two numbers that multiply to \(-120\) and add to \(-3\). The numbers are \(8\) and \(-15\) because:
\[8 \cdot (-15) = -120\]
\[8 + (-15) = -3\]
3. We can now rewrite the middle term of the quadratic expression using these numbers:
\[
8x^2 + 8x - 15x - 15
\]
4. Now, group the terms:
\[
(8x^2 + 8x) + (-15x - 15)
\]
5. Factor out the common factors from each group:
\[
8x(x + 1) - 15(x + 1)
\]
6. Now, we can factor out the common binomial factor \((x + 1)\):
\[
(8x - 15)(x + 1)
\]
Thus, the factored form of the expression \(8x^2 - 3x - 15\) is:
\[
(8x - 15)(x + 1)
\]
Here, \(a = 8\), \(b = -3\), and \(c = -15\).
1. First, calculate \(a \cdot c = 8 \cdot (-15) = -120\).
2. Next, we need to find two numbers that multiply to \(-120\) and add to \(-3\). The numbers are \(8\) and \(-15\) because:
\[8 \cdot (-15) = -120\]
\[8 + (-15) = -3\]
3. We can now rewrite the middle term of the quadratic expression using these numbers:
\[
8x^2 + 8x - 15x - 15
\]
4. Now, group the terms:
\[
(8x^2 + 8x) + (-15x - 15)
\]
5. Factor out the common factors from each group:
\[
8x(x + 1) - 15(x + 1)
\]
6. Now, we can factor out the common binomial factor \((x + 1)\):
\[
(8x - 15)(x + 1)
\]
Thus, the factored form of the expression \(8x^2 - 3x - 15\) is:
\[
(8x - 15)(x + 1)
\]
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