Question

week 1: target d - a-sse.2: interpret the structure of expressions.

1. determine whether each equation is true for all values of x.

select yes or no for each equation.
(table with equations: a: 8^x = 2^(3x), b: 9^x = 3^(2x), c: 16^x = 2^(4x), d: 5^(2x) = 2^(5x))

1. select the expression that is equivalent to n² - 121m²

a) n² - 22mn + 121m²
b) (n - 11m)(n + 11m)
c) (n + 11m)²
d) (n - 11m)²

1. select the expression that is equivalent to 25x² - 49.

a) (5x - 7)²
b) (5x + 7)²
c) (5x - 7)(5x + 7)
d) 25x² - 70x - 49

1. select the expression that is equivalent to 2(x - 3)².

a) 2(x² - 9)
b) 4x² - 12
c) 4x² + 12
d) 2(x² + 6x + 9)
e) 2x² - 12x + 18

1. use the equation below to answer the question.

f(x) = (x - 7)(x + 2)
which equivalent equation is correctly matched with a key feature of the graph of the function it represents?
a) f(x) = x² - 5x - 14 highlights that the x intercept is -5.
b) f(x) = x² - 5x - 14 highlights that the x intercept is -14.
c) f(x) = x² - 5x - 14 highlights that the y intercept is -5.
d) f(x) = x² - 5x - 14 highlights that the y intercept is -14.

Explanation:

Step1: Rewrite bases as powers

Equation A:

$6^x$ cannot be rewritten as a power of 2, so $6^x
eq 2^{3x}$ for all $x$.

Equation B:

Rewrite $9$ as $3^2$:
$9^x = (3^2)^x = 3^{2x}
eq 3^{3x}$ for all $x$.

Equation C:

Rewrite $16$ as $2^4$:
$16^x = (2^4)^x = 2^{4x}$ which matches $2^{4x}$.

Equation D:

$5^{3x}$ cannot be rewritten as a power of 2, so $5^{3x}
eq 2^{5x}$ for all $x$.

Step2: Factor difference of squares

For $n^2 - 121m^2$, recognize it as $a^2 - b^2$ where $a=n$, $b=11m$.
The formula is $a^2 - b^2=(a-b)(a+b)$, so:
$n^2 - 121m^2=(n-11m)(n+11m)$

Step3: Factor difference of squares

For $25x^2 - 49$, recognize it as $a^2 - b^2$ where $a=5x$, $b=7$.
Apply the difference of squares formula:
$25x^2 - 49=(5x-7)(5x+7)$

Step4: Expand squared binomial first

First expand $(x-3)^2$ using $(a-b)^2=a^2-2ab+b^2$:
$(x-3)^2=x^2-6x+9$
Multiply by 2:
$2(x^2-6x+9)=2x^2-12x+18$

Step5: Expand and analyze intercepts

First expand $f(x)=(x-7)(x+2)$:
$f(x)=x^2+2x-7x-14=x^2-5x-14$

• x-intercepts: Set $f(x)=0$, so $(x-7)(x+2)=0$, intercepts at $x=7$ and $x=-2$.
• y-intercept: Set $x=0$, $f(0)=0-0-14=-14$.

Answer:

1. A: No, B: No, C: Yes, D: No
2. B) $(n - 11m) (n + 11m)$
3. C) $(5x - 7) (5x + 7)$
4. E) $2x^2 - 12x + 18$
5. D) $f(x) = x^2 - 5x - 14$ highlights that the y intercept is -14.