If $ x^2 + 7x + 8 = 0 $, then find the value of $ {4x}/{x^2 - 5x + 8} $.

Question

Accepted Answer

Given the quadratic equation:
$$
x^2 + 7x + 8 = 0
$$
We need to find the value of:
$$
\frac{4x}{x^2 - 5x + 8}
$$

Step 1: Express $x^2$ from the given equation:
$$
x^2 = -7x - 8
$$

Step 2: Substitute $x^2$ in the denominator:
$$
x^2 - 5x + 8 = (-7x - 8) - 5x + 8 = -7x - 8 - 5x + 8 = -12x
$$

Step 3: So the expression becomes:
$$
\frac{4x}{-12x} = \frac{4x}{-12x} = -\frac{1}{3}
$$

However, we must be careful about the domain. Since $x$ is a root of the quadratic, $x \neq 0$ (otherwise the equation would be 8=0, which is false), so division by $x$ is valid.

Step 4: Re-examining the denominator substitution:
$$
x^2 - 5x + 8 = (-7x - 8) - 5x + 8 = -7x - 8 - 5x + 8 = -12x
$$

Therefore:
$$
\frac{4x}{x^2 - 5x + 8} = \frac{4x}{-12x} = -\frac{1}{3}
$$

Step 5: Check the options:
- Option 3 is $-\frac{1}{3}$, which matches our result.

Hence, the correct answer is $-\frac{1}{3}$.

Answer

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