Question

question
write two numbers that multiply to the value on top and add to the value on bottom.
-65
×
+
-8
answer attempt 1 out of 2
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Explanation:

Step1: Define variables

Let the two numbers be $x$ and $y$. We know:
$$x \times y = -65$$
$$x + y = -8$$

Step2: Rewrite and substitute

From the second equation: $y = -8 - x$. Substitute into the first equation:
$$x(-8 - x) = -65$$
$$-8x - x^2 = -65$$
$$x^2 + 8x - 65 = 0$$

Step3: Solve quadratic equation

Use quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ where $a=1$, $b=8$, $c=-65$:
$$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-65)}}{2(1)}$$
$$x = \frac{-8 \pm \sqrt{64 + 260}}{2}$$
$$x = \frac{-8 \pm \sqrt{324}}{2}$$
$$x = \frac{-8 \pm 18}{2}$$

Step4: Calculate two values

First value: $x = \frac{-8 + 18}{2} = \frac{10}{2} = 5$
Second value: $x = \frac{-8 - 18}{2} = \frac{-26}{2} = -13$
Verify: $5 \times (-13) = -65$, $5 + (-13) = -8$

Answer:

$5$ and $-13$