Question

find the error if a student multiplied $(-3x - 4)^2$ and got an answer of $-9x^2 + 24x + 16$.

write the correct product.

Explanation:

Step1: Recall the square of a binomial formula

The formula for \((a + b)^2\) is \(a^2 + 2ab + b^2\), and for \((a - b)^2\) is \(a^2 - 2ab + b^2\). Here, we can rewrite \((-3x - 4)^2\) as \([- (3x + 4)]^2=(3x + 4)^2\) (since \((-a)^2 = a^2\)). Also, we can use the formula \((a + b)^2=a^2 + 2ab + b^2\) where \(a=-3x\) and \(b = - 4\).

Step2: Analyze the student's error

The student's answer is \(-9x^2+24x + 16\). Let's expand \((-3x - 4)^2\) correctly. Using \((a + b)^2=a^2+2ab + b^2\) with \(a=-3x\) and \(b=-4\):

• \(a^2=(-3x)^2 = 9x^2\) (the student got \(-9x^2\), which is wrong. The square of a negative number is positive, so \((-3x)^2=9x^2\), not \(-9x^2\))
• \(2ab=2\times(-3x)\times(-4)=24x\) (this part is correct)
• \(b^2=(-4)^2 = 16\) (this part is correct)

Step3: Find the correct product

Now, expand \((-3x - 4)^2\) using the formula \((a + b)^2=a^2+2ab + b^2\) with \(a=-3x\) and \(b = - 4\):
\[

$$\begin{align*} (-3x-4)^2&=(-3x)^2+2\times(-3x)\times(-4)+(-4)^2\\ &=9x^2 + 24x+16 \end{align*}$$

\]
Or we can also expand it as \((-3x - 4)(-3x - 4)\):
\[

$$\begin{align*} (-3x)\times(-3x)+(-3x)\times(-4)+(-4)\times(-3x)+(-4)\times(-4)&=9x^2 + 12x+12x + 16\\ &=9x^2+24x + 16 \end{align*}$$

\]

Answer:

Error:

The student incorrectly calculated \((-3x)^2\) as \(-9x^2\) instead of \(9x^2\) (the square of a negative term \(-3x\) should be positive \(9x^2\), not negative \(-9x^2\)).

Correct Product:

\(9x^2 + 24x + 16\)