Question

find the factors of the above equation.

\\(x^2 + 4x - 480 = 0\\)

mark all correct answers

a. \\((x + 24)\\)
b. \\((x - 24)\\)
c. \\((x + 20)\\)
d. \\((x - 20)\\)

Explanation:

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<reasoning>

Identify the quadratic equation to factor

Using the Quadratic Equations knowledge point
\[
x^2 + 4x - 480 = 0
\]

Find two numbers that multiply to -480 and add to 4

We need to find two integers, \(a\) and \(b\), such that:
\[

$$\begin{aligned} a \cdot b &= -480 \\ a + b &= 4 \end{aligned}$$

\]
Since the product is negative, one number is positive and the other is negative. Since their sum is positive, the number with the larger absolute value is positive.
Let's test factors of \(480\) that are close to \(\sqrt{480} \approx 21.9\):
\[

$$\begin{aligned} 24 \cdot (-20) &= -480 \\ 24 + (-20) &= 4 \end{aligned}$$

\]
Thus, the two numbers are \(24\) and \(-20\).

Write the factored form of the equation

Using the Quadratic Equations knowledge point
\[
(x + 24)(x - 20) = 0
\]
The factors of the equation are \((x + 24)\) and \((x - 20)\).

Match the factors with the given options

The factors we found are:

• \((x + 24)\), which corresponds to option a.
• \((x - 20)\), which corresponds to option d.

</reasoning>

<answer>
<mcq-correct>a. \((x + 24)\)</mcq-correct>
<mcq-option>b. \((x - 24)\)</mcq-option>
<mcq-option>c. \((x + 20)\)</mcq-option>
<mcq-correct>d. \((x - 20)\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the quadratic equation to factor

Using the Quadratic Equations knowledge point
\[
x^2 + 4x - 480 = 0
\]

Find two numbers that multiply to -480 and add to 4

We need to find two integers, \(a\) and \(b\), such that:
\[

$$\begin{aligned} a \cdot b &= -480 \\ a + b &= 4 \end{aligned}$$

\]
Since the product is negative, one number is positive and the other is negative. Since their sum is positive, the number with the larger absolute value is positive.
Let's test factors of \(480\) that are close to \(\sqrt{480} \approx 21.9\):
\[

$$\begin{aligned} 24 \cdot (-20) &= -480 \\ 24 + (-20) &= 4 \end{aligned}$$

\]
Thus, the two numbers are \(24\) and \(-20\).

Write the factored form of the equation

Using the Quadratic Equations knowledge point
\[
(x + 24)(x - 20) = 0
\]
The factors of the equation are \((x + 24)\) and \((x - 20)\).

Match the factors with the given options

The factors we found are:

• \((x + 24)\), which corresponds to option a.
• \((x - 20)\), which corresponds to option d.

</reasoning>

<answer>
<mcq-correct>a. \((x + 24)\)</mcq-correct>
<mcq-option>b. \((x - 24)\)</mcq-option>
<mcq-option>c. \((x + 20)\)</mcq-option>
<mcq-correct>d. \((x - 20)\)</mcq-correct>
</answer>

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