Question

1. $14m^4 + 152m^2 + 120$
2. $2x^4 - 23x^2 + 45$
3. johnny is studying the function, f(x) that is shown in the graph below. he claims he can transform the function to include the ordered par (2,2). write the function that would support his claim:
4. johnny is studying the function, f(x) that is shown in the graph below. he claims he can transform the function to include the ordered par (-4,-3). write the function that would support his claim:

circle the equation that has no real solutions

1. $x^2 + 2x - 35 = 0$

$3x^2 - 6x + 2 = 0$
$2x^2 - 3x + 5 = 0$
$x^2 + 5x - 2 = 0$

1. $a^2 + 3a - 4 = 0$

$a^2 - 2a + 1 = 0$
$3a^2 + 5a - 2 = 0$
$2a^2 - 3a + 6 = 0$
for each equation given, state if it has no solutions, one solution, or infinitely many solutions

1. a. $sqrt{x} - 10 = -7$

b. $3sqrt{x} + 12 = 3$
c. $\frac{2}{x - 1} = \frac{3}{x + 2}$

1. a. $sqrt{x} + 2 = -5$

b. $4sqrt{x} - 8 = 24$
c. $2(3x - 1) = 4x - 2 + 2x$

Explanation:

Problem 23

Step1: Substitute $u=m^2$

Let $u=m^2$, rewrite the expression:
$14u^2 + 152u + 120$

Step2: Factor out GCF

Factor out 2:
$2(7u^2 + 76u + 60)$

Step3: Factor quadratic in $u$

Find two numbers: $70$ and $6$. Split and factor:
$2(7u(u+10)+6(u+10))=2(7u+6)(u+10)$

Step4: Substitute back $u=m^2$

Replace $u$ with $m^2$:
$2(7m^2+6)(m^2+10)$

Step1: Substitute $u=x^2$

Let $u=x^2$, rewrite the expression:
$2u^2 -23u +45$

Step2: Factor quadratic in $u$

Find two numbers: $-18$ and $-5$. Split and factor:
$2u(u-9)-5(u-9)=(2u-5)(u-9)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$, factor difference of squares:
$(2x^2-5)(x-3)(x+3)$

Step1: Identify parent function

The graph is a downward parabola with vertex $(0,2)$:
$f(x)=-2x^2+2$

Step2: Apply horizontal shift right 2

To include $(2,2)$, shift right 2 units:
$f(x-2)=-2(x-2)^2+2$

Answer:

$2(7m^2+6)(m^2+10)$

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Problem 24