Question

solve 5x^2+9x-2=0 by factoring. Enter your answers below
x=
or x=

Answer 1

To solve the quadratic equation \(5x^2 + 9x - 2 = 0\) by factoring, we need to express it in the factored form.

1. First, we multiply the coefficient of \(x^2\) (which is 5) with the constant term (which is -2):
\[
5 \cdot -2 = -10
\]

2. Now, we need to find two numbers that multiply to -10 and add up to 9 (the coefficient of \(x\)). The numbers that meet these criteria are 10 and -1 (since \(10 \cdot -1 = -10\) and \(10 + (-1) = 9\)).

3. We can now split the middle term using these two numbers:
\[
5x^2 + 10x - 1x - 2 = 0
\]

4. Next, we group the terms:
\[
(5x^2 + 10x) + (-1x - 2) = 0
\]

5. Factor out the common factors from each group:
\[
5x(x + 2) - 1(x + 2) = 0
\]

6. Now, factor out the common binomial \((x + 2)\):
\[
(5x - 1)(x + 2) = 0
\]

7. Set each factor to zero and solve for \(x\):
\[
5x - 1 = 0 \quad \text{or} \quad x + 2 = 0
\]

For the first factor:
\[
5x = 1 \implies x = \frac{1}{5}
\]

For the second factor:
\[
x = -2
\]

So the solutions are:
\[
x = \frac{1}{5} \quad \text{or} \quad x = -2
\]

Thus, the final answers are:
\[
x = \frac{1}{5} \quad \text{or} \quad x = -2
\]