Question

4x^2 - 17x + 15 9th grade algebra solve the trinomial

Answer 1

To solve the trinomial \(4x^2 - 17x + 15\), we can factor it. We need to find two numbers that multiply to \(4 \cdot 15 = 60\) (the product of the leading coefficient and the constant) and add to \(-17\) (the coefficient of \(x\)).

Let's find the pair of numbers:

The pairs of factors of 60 are:
- (1, 60)
- (2, 30)
- (3, 20)
- (4, 15)
- (5, 12)
- (6, 10)

We look for a pair that adds to \(-17\):
- The pair \(-12\) and \(-5\) works since \(-12 + (-5) = -17\).

Now we can rewrite the middle term of the trinomial using \(-12\) and \(-5\):

\[
4x^2 - 12x - 5x + 15
\]

Next, we can group the terms:

\[
(4x^2 - 12x) + (-5x + 15)
\]

Factoring each group gives us:

\[
4x(x - 3) - 5(x - 3)
\]

Now we can factor out \((x - 3)\):

\[
(4x - 5)(x - 3)
\]

So the factored form of the trinomial is:

\[
(4x - 5)(x - 3) = 0
\]

Now, we can set each factor equal to zero to find the solutions:

1. \(4x - 5 = 0\)
\[
4x = 5 \\
x = \frac{5}{4}
\]

2. \(x - 3 = 0\)
\[
x = 3
\]

Thus, the solutions to the equation \(4x^2 - 17x + 15 = 0\) are:

\[
x = \frac{5}{4} \quad \text{and} \quad x = 3
\]