Solving Quadratic Equations Using Common Factors Practice

Question

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Question
Karim is solving the quadratic equation x2+5x−24=0
by grouping. His work is shown below. Determine the first step at which Karim makes an error.

Step 1: x2+8x+(−3x)−24=0


Step 2: (x2+8x)+(−3x−24)=0


Step 3: x(x+8)+3(x+8)=0


Step 4: (x+8)(x+3)=0


Step 5: x+8=0
or x+3=0


x=−8
or x=−3


(1 point)
Karim's first mistake occurs at Step
.
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Answer 1

Let's examine the steps that Karim took to solve the quadratic equation \(x^2 + 5x - 24 = 0\) by grouping:

1. Step 1: \(x^2 + 8x + (-3x) - 24 = 0\)

   • Here, Karim incorrectly chose \(8\) and \(-3\) to split the middle term \(5\). The correct factors that sum up to \(5\) and multiply to \(-24\) should be \(8\) and \(-3\), so this step seems acceptable at first glance.

2. Step 2: \((x^2 + 8x) + (-3x - 24) = 0\)

   • This step correctly groups the terms.

3. Step 3: \(x(x + 8) + 3(x + 8) = 0\)

   • Here, he factors out \(x\) from \(x^2 + 8x\) (correct) but he mistakenly factors out \(3\) from \(-3x - 24\) instead of \(-3\) as it should be factoring out \(-3\) from that term.

4. Step 4: \((x + 8)(x + 3) = 0\)

   • This step results from a misapplication or misunderstanding of step 3.

5. Step 5: \(x + 8 = 0\) or \(x + 3 = 0\)

   • This leads to the roots \(x = -8\) or \(x = -3\).

Karim's first error occurs in Step 3, where he incorrectly factored \(-3x - 24\). Instead, he should have factored out \(-3\), leading to correct factors.

Therefore, the correct answer is that Karim's first mistake occurs at Step 3.