Question

b
chris needs to fill
in the blank to
create a perfect
square trinomial.
which could be
the missing term?
$4x^2 + \underline{\quad\quad} + 25$

a. $10x$ or $-10x$
b. $20x$ or $-20x$
c. $40x$ or $-40x$

d
ally multiplied
$(7x + 5)$ by a
second factor and
got a product of
$49x^2 + 70x + 25$.
find the second
factor.

a. $(7x + 10)$
b. $(7x - 5)$
c. $(7x + 5)$

Explanation:

Problem B

Step1: Recall perfect square trinomial formula

A perfect square trinomial is of the form \((ax \pm b)^2=a^{2}x^{2}\pm2abx + b^{2}\). For the given trinomial \(4x^{2}+\underline{\quad}+25\), we can see that \(a^{2}x^{2} = 4x^{2}\), so \(a^{2}=4\), which means \(a = 2\) (since \(a\) is a real number, we take the positive root for the coefficient). And \(b^{2}=25\), so \(b=\pm5\).

Step2: Calculate the middle term

Using the formula for the middle term \(\pm2abx\), substitute \(a = 2\) and \(b=\pm5\). Then \(2ab=2\times2\times5 = 20\) or \(2ab=2\times2\times(- 5)=- 20\). So the middle term should be \(20x\) or \(-20x\).

Step1: Recall the formula for squaring a binomial

The formula for \((a + b)^{2}=a^{2}+2ab + b^{2}\). We are given that one factor is \((7x + 5)\) and the product is \(49x^{2}+70x + 25\). Let's check the product of \((7x + 5)\) with each option.

Step2: Check option a

Multiply \((7x + 5)(7x+10)=49x^{2}+70x + 35x+50=49x^{2}+105x + 50
eq49x^{2}+70x + 25\).

Step3: Check option b

Multiply \((7x + 5)(7x - 5)=49x^{2}-25
eq49x^{2}+70x + 25\).

Step4: Check option c

Multiply \((7x + 5)(7x + 5)=(7x + 5)^{2}=49x^{2}+70x + 25\), which matches the given product.

Answer:

b. \(20x\) or \(-20x\)

Problem D