Question

4 multiple choice 20 points
which expression is equivalent to $x^2 + 10x + 25$?
○ (x+2)(x+12)
○ (x+1)(x+25)
○ (x+10)(x+1)
○ $(x + 5)^2$

5 multiple choice 20 points
what is the factored form of:
$6x^2 + 12x$?
○ (2x)(3x+6)
○ 6x(x+2)
○ 3x(2x+6)
○ $2\left(x^2 + 2x\
ight)$

Explanation:

Question 4

Step1: Recall perfect square formula

The perfect square trinomial formula is \((a + b)^2 = a^2 + 2ab + b^2\). For \(x^2+10x + 25\), we have \(a=x\), and we need to find \(b\) such that \(2ab = 10x\) and \(b^2=25\).

Step2: Solve for \(b\)

From \(b^2 = 25\), we get \(b = 5\) (since \(2ab=2\times x\times5 = 10x\) which matches the middle term). So \(x^2 + 10x+25=(x + 5)^2\). We can also expand the other options:

• \((x + 2)(x+12)=x^2+14x + 24

eq x^2+10x + 25\)

• \((x + 1)(x+25)=x^2+26x + 25

eq x^2+10x + 25\)

• \((x + 10)(x+1)=x^2+11x + 10

eq x^2+10x + 25\)

Step1: Find the GCF

The greatest common factor (GCF) of \(6x^2\) and \(12x\) is \(6x\) (since \(6x\times x=6x^2\) and \(6x\times2 = 12x\)).

Step2: Factor out the GCF

Using the distributive property \(ab+ac=a(b + c)\), where \(a = 6x\), \(b=x\), and \(c = 2\), we get \(6x^2+12x=6x(x + 2)\). Let's check the other options:

• \((2x)(3x + 6)=6x^2+12x\), but it's not fully factored (3x + 6 still has a GCF of 3).
• \(3x(2x + 6)=6x^2+18x

eq6x^2+12x\)

• \(2(x^2+2x)=2x^2+4x

eq6x^2+12x\)

Answer:

\(\boldsymbol{(x + 5)^2}\) (the fourth option)

Question 5