Question

#1-5
name:
unit 4: solving quadratic equations
date:
per:
homework 12: solving nonlinear systems graphically
this is a 2-page document!
solve each system by graphing. be sure to identify the solution(s).

1. $\

$$\begin{cases} y = 4x + 5 \\\\ y = x^2 + 8x + 9 \\end{cases}$$

$

1. $\

$$\begin{cases} y = -x^2 - 4x + 2 \\\\ y = x^2 + 8x + 12 \\end{cases}$$

$

1. $\

$$\begin{cases} y = x^2 + 12x + 26 \\\\ y = -x^2 - 4x - 6 \\end{cases}$$

$

1. $\

$$\begin{cases} y = -x^2 + 10x - 28 \\\\ x - y = 7 \\end{cases}$$

$

1. $\

$$\begin{cases} y = x^2 - 2x - 7 \\\\ y = -3x - 5 \\end{cases}$$

$

1. $\

$$\begin{cases} y = x^2 - 6x + 10 \\\\ y = -2x^2 + 4x \\end{cases}$$

$

Explanation:

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Problem 1

Step1: Set equations equal

$4x + 5 = x^2 + 8x + 9$

Step2: Rearrange to standard quadratic

$x^2 + 4x + 4 = 0$

Step3: Factor quadratic

$(x+2)^2 = 0$

Step4: Solve for x, find y

$x=-2$, $y=4(-2)+5=-3$

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Problem 2

Step1: Set equations equal

$-x^2 -4x +2 = x^2 +8x +12$

Step2: Rearrange to standard quadratic

$2x^2 +12x +10=0$

Step3: Simplify quadratic

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

Step4: Factor and solve for x

$(x+1)(x+5)=0$ → $x=-1, x=-5$

Step5: Find corresponding y-values

For $x=-1$: $y=-(-1)^2-4(-1)+2=5$
For $x=-5$: $y=-(-5)^2-4(-5)+2=-3$

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Problem 3

Step1: Set equations equal

$x^2 +12x +26 = -x^2 -4x -6$

Step2: Rearrange to standard quadratic

$2x^2 +16x +32=0$

Step3: Simplify quadratic

$x^2 +8x +16=0$

Step4: Factor and solve for x

$(x+4)^2=0$ → $x=-4$

Step5: Find corresponding y-value

$y=-(-4)^2-4(-4)-6=-6$

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Problem 4

Step1: Rearrange linear equation

$y=x-7$

Step2: Set equations equal

$x-7 = -x^2 +10x -28$

Step3: Rearrange to standard quadratic

$x^2 -9x +21=0$

Step4: Check discriminant

$\Delta = (-9)^2 -4(1)(21)=81-84=-3 <0$

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Problem 5

Step1: Set equations equal

$x^2 -2x -7 = -3x -5$

Step2: Rearrange to standard quadratic

$x^2 +x -2=0$

Step3: Factor and solve for x

$(x+2)(x-1)=0$ → $x=-2, x=1$

Step4: Find corresponding y-values

For $x=-2$: $y=-3(-2)-5=1$
For $x=1$: $y=-3(1)-5=-8$

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Problem 6

Step1: Set equations equal

$x^2 -6x +10 = -2x^2 +4x$

Step2: Rearrange to standard quadratic

$3x^2 -10x +10=0$

Step3: Check discriminant

$\Delta = (-10)^2 -4(3)(10)=100-120=-20 <0$

Answer:

1. Solution: $(-2, -3)$
2. Solutions: $(-1, 5)$ and $(-5, -3)$
3. Solution: $(-4, -6)$
4. No real solutions
5. Solutions: $(-2, 1)$ and $(1, -8)$
6. No real solutions