Question

1. which expressions are equivalent to $9x^2 - 49$?

select two answer choices.
a. $(3x - 7)^2$
b. $(3x + 7)^2$
c. $(3x)^2 - (7)^2$
d. $(3x - 7)(3x + 7)$
e. $(3x - 7)(3x - 7)$

1. which expressions are equivalent to $16x^2 - 1$?

select two answer choices.
a. $(4x - 1)^2$
b. $(4x + 1)^2$
c. $(4x)^2 - (1)^2$
d. $(4x - 1)(4x + 1)$
e. $(4x - 1)(4x - 1)$

1. which expressions are equivalent to $x^2 + 5x - 14$?

select two answer choices.
a. $(x - 2)(x - 7)$
b. $(x + 7)(x - 2)$
c. $(x + 5)(x - 14)$
d. $(x + 2.5)^2 - 20.25$
e. $(x + 2.5)^2 + 20.25$

Explanation:

Problem 1

Step1: Recall Difference of Squares

The expression \(9x^2 - 49\) is a difference of squares, since \(9x^2=(3x)^2\) and \(49 = 7^2\). The formula for difference of squares is \(a^2 - b^2=(a - b)(a + b)\).

Step2: Analyze Option C

Option C is \((3x)^2-(7)^2\), which is the same as \(9x^2 - 49\) by substituting \(a = 3x\) and \(b = 7\) into the difference of squares form.

Step3: Analyze Option D

Option D is \((3x - 7)(3x + 7)\), which is the factored form of \(9x^2 - 49\) using the difference of squares formula (\(a^2 - b^2=(a - b)(a + b)\) with \(a = 3x\) and \(b = 7\)).

Step4: Eliminate Other Options

• Option A: \((3x - 7)^2=(3x - 7)(3x - 7)=9x^2-42x + 49

eq9x^2 - 49\)

• Option B: \((3x + 7)^2=(3x + 7)(3x + 7)=9x^2 + 42x + 49

eq9x^2 - 49\)

• Option E: \((3x - 7)(3x - 7)=(3x - 7)^2=9x^2-42x + 49

eq9x^2 - 49\)

Step1: Recall Difference of Squares

The expression \(16x^2-1\) is a difference of squares, since \(16x^2=(4x)^2\) and \(1 = 1^2\). The formula for difference of squares is \(a^2 - b^2=(a - b)(a + b)\).

Step2: Analyze Option C

Option C is \((4x)^2-(1)^2\), which is the same as \(16x^2 - 1\) by substituting \(a = 4x\) and \(b = 1\) into the difference of squares form.

Step3: Analyze Option D

Option D is \((4x - 1)(4x + 1)\), which is the factored form of \(16x^2 - 1\) using the difference of squares formula (\(a^2 - b^2=(a - b)(a + b)\) with \(a = 4x\) and \(b = 1\)).

Step4: Eliminate Other Options

• Option A: \((4x - 1)^2=(4x - 1)(4x - 1)=16x^2-8x + 1

eq16x^2 - 1\)

• Option B: \((4x + 1)^2=(4x + 1)(4x + 1)=16x^2 + 8x + 1

eq16x^2 - 1\)

• Option E: \((4x - 1)(4x - 1)=(4x - 1)^2=16x^2-8x + 1

eq16x^2 - 1\)

Step1: Factor the Quadratic (Option B)

We factor \(x^2 + 5x - 14\). We need two numbers that multiply to \(- 14\) and add to \(5\). The numbers are \(7\) and \(-2\). So, \(x^2+5x - 14=(x + 7)(x - 2)\), which is Option B.

Step2: Complete the Square (Option D)

To complete the square for \(x^2+5x - 14\), we take the coefficient of \(x\), which is \(5\), divide by \(2\) to get \(\frac{5}{2}=2.5\), and square it to get \((2.5)^2 = 6.25\).
We rewrite the quadratic as:

$$
LATEXBLOCK0
$$

Which is Option D.

Step3: Eliminate Other Options

• Option A: \((x - 2)(x - 7)=x^2-9x + 14

eq x^2+5x - 14\)

• Option C: \((x + 5)(x - 14)=x^2-9x - 70

eq x^2+5x - 14\)

• Option E: \((x + 2.5)^2+20.25=x^2+5x + 6.25+20.25=x^2+5x + 26.5

eq x^2+5x - 14\)

Answer:

C. \((3x)^2-(7)^2\), D. \((3x - 7)(3x + 7)\)

Problem 2