Question
Factor this trinomial completely
-12x^2 + 45x + 12
-12x^2 + 45x + 12
Answers
GPT-4o mini
To factor the trinomial \(-12x^2 + 45x + 12\) completely, we can start by finding the greatest common factor (GCF) of the coefficients.
1. The coefficients are \(-12\), \(45\), and \(12\). The GCF is \(3\) (we'll account for the negative later), so we will factor out \(-3\):
\[
-3(4x^2 - 15x - 4)
\]
Now we will focus on factoring the quadratic \(4x^2 - 15x - 4\).
2. We need to find two numbers that multiply to \(4 \cdot (-4) = -16\) (the product of the leading coefficient and the constant term) and add up to \(-15\) (the coefficient of the middle term).
The numbers that work are \(-16\) and \(1\).
3. Now we can rewrite the middle term using these numbers:
\[
4x^2 - 16x + 1x - 4
\]
4. Now group the terms:
\[
(4x^2 - 16x) + (1x - 4)
\]
5. Factor each group:
\[
4x(x - 4) + 1(x - 4)
\]
6. Now, we can factor out the common factor \((x - 4)\):
\[
(4x + 1)(x - 4)
\]
7. Now incorporating the \(-3\) we factored out earlier, we have:
\[
-3(4x + 1)(x - 4)
\]
Thus, the completely factored form of the original trinomial \(-12x^2 + 45x + 12\) is:
\[
-3(4x + 1)(x - 4)
\]
1. The coefficients are \(-12\), \(45\), and \(12\). The GCF is \(3\) (we'll account for the negative later), so we will factor out \(-3\):
\[
-3(4x^2 - 15x - 4)
\]
Now we will focus on factoring the quadratic \(4x^2 - 15x - 4\).
2. We need to find two numbers that multiply to \(4 \cdot (-4) = -16\) (the product of the leading coefficient and the constant term) and add up to \(-15\) (the coefficient of the middle term).
The numbers that work are \(-16\) and \(1\).
3. Now we can rewrite the middle term using these numbers:
\[
4x^2 - 16x + 1x - 4
\]
4. Now group the terms:
\[
(4x^2 - 16x) + (1x - 4)
\]
5. Factor each group:
\[
4x(x - 4) + 1(x - 4)
\]
6. Now, we can factor out the common factor \((x - 4)\):
\[
(4x + 1)(x - 4)
\]
7. Now incorporating the \(-3\) we factored out earlier, we have:
\[
-3(4x + 1)(x - 4)
\]
Thus, the completely factored form of the original trinomial \(-12x^2 + 45x + 12\) is:
\[
-3(4x + 1)(x - 4)
\]