Question

1. which function is equivalent to ( q(x) = 9x^2 - 24x + 16 )?

• ( q(x) = (3x + 4)^2 )
• ( q(x) = (9x + 4)(x + 4) )
• ( q(x) = (9x - 4)(x - 4) )
• ( q(x) = (3x - 4)^2 )

Explanation:

Step1: Recall the perfect square formula

The perfect square formula is \((a - b)^2 = a^2 - 2ab + b^2\) and \((a + b)^2 = a^2 + 2ab + b^2\). We will expand each option and check which one matches \(q(x)=9x^2 - 24x + 16\).

Step2: Expand option 1: \((3x + 4)^2\)

Using the formula \((a + b)^2=a^2 + 2ab + b^2\), where \(a = 3x\) and \(b = 4\).
\((3x+4)^2=(3x)^2+2\times(3x)\times4 + 4^2=9x^2+24x + 16\). This does not match \(9x^2-24x + 16\).

Step3: Expand option 2: \((9x + 4)(x + 4)\)

Using the distributive property (FOIL method): \((9x+4)(x + 4)=9x\times x+9x\times4+4\times x + 4\times4=9x^2+36x+4x + 16=9x^2+40x + 16\). This does not match.

Step4: Expand option 3: \((9x - 4)(x - 4)\)

Using the distributive property (FOIL method): \((9x-4)(x - 4)=9x\times x-9x\times4-4\times x+4\times4=9x^2-36x-4x + 16=9x^2-40x + 16\). This does not match.

Step5: Expand option 4: \((3x - 4)^2\)

Using the formula \((a - b)^2=a^2-2ab + b^2\), where \(a = 3x\) and \(b = 4\).
\((3x - 4)^2=(3x)^2-2\times(3x)\times4+4^2=9x^2-24x + 16\). This matches the given function \(q(x)\).

Answer:

D. \(q(x)=(3x - 4)^2\) (assuming the last option is D, if the options are labeled as A, B, C, D with the last one being D. If the original options are labeled differently, adjust the label accordingly. Since the last option is \(q(x)=(3x - 4)^2\), the answer is the option with \(q(x)=(3x - 4)^2\))