Question

1. $x^{2}+4x-21=0$
2. $n^{2}-3n+2=0$
3. $4v^{2}+5v+1=0$
4. $2b^{2}+3b=0$
5. $m^{2}=12-4m$
6. $x^{2}=-42-13x$
7. $4n^{2}=25$
8. $5p^{2}=2p$

Explanation:

Problem 9: $x^2+4x-21=0$

Step1: Factor the quadratic

Find two numbers that multiply to $-21$ and add to $4$: $7$ and $-3$.
$(x+7)(x-3)=0$

Step2: Solve for $x$

Set each factor to 0:
$x+7=0 \implies x=-7$; $x-3=0 \implies x=3$

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Problem10: $n^2-3n+2=0$

Step1: Factor the quadratic

Find two numbers that multiply to $2$ and add to $-3$: $-1$ and $-2$.
$(n-1)(n-2)=0$

Step2: Solve for $n$

Set each factor to 0:
$n-1=0 \implies n=1$; $n-2=0 \implies n=2$

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Problem11: $4v^2+5v+1=0$

Step1: Use quadratic formula

For $av^2+bv+c=0$, $v=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=4, b=5, c=1$.

Step2: Calculate values

$\Delta=5^2-4(4)(1)=25-16=9$
$v=\frac{-5\pm\sqrt{9}}{8}=\frac{-5\pm3}{8}$
Simplify: $v=\frac{-2}{8}=-\frac{1}{4}$ or $v=\frac{-8}{8}=-1$

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Problem12: $2b^2+3b=0$

Step1: Factor out common term

$b(2b+3)=0$

Step2: Solve for $b$

Set each factor to 0:
$b=0$; $2b+3=0 \implies b=-\frac{3}{2}$

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Problem13: $m^2=12-4m$

Step1: Rearrange to standard form

$m^2+4m-12=0$

Step2: Factor the quadratic

Find two numbers that multiply to $-12$ and add to $4$: $6$ and $-2$.
$(m+6)(m-2)=0$

Step3: Solve for $m$

$m+6=0 \implies m=-6$; $m-2=0 \implies m=2$

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Problem14: $x^2=-42-13x$

Step1: Rearrange to standard form

$x^2+13x+42=0$

Step2: Factor the quadratic

Find two numbers that multiply to $42$ and add to $13$: $6$ and $7$.
$(x+6)(x+7)=0$

Step3: Solve for $x$

$x+6=0 \implies x=-6$; $x+7=0 \implies x=-7$

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Problem15: $4n^2=25$

Step1: Isolate $n^2$

$n^2=\frac{25}{4}$

Step2: Take square root

$n=\pm\sqrt{\frac{25}{4}}=\pm\frac{5}{2}$

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Problem16: $5p^2=2p$

Step1: Rearrange to standard form

$5p^2-2p=0$

Step2: Factor out common term

$p(5p-2)=0$

Step3: Solve for $p$

$p=0$; $5p-2=0 \implies p=\frac{2}{5}$

Answer:

1. $x=3$ or $x=-7$
2. $n=1$ or $n=2$
3. $v=\frac{-5+\sqrt{9}}{8}$ or $v=\frac{-5-\sqrt{9}}{8}$ (simplified: $v=-\frac{1}{4}$ or $v=-1$)
4. $b=0$ or $b=-\frac{3}{2}$
5. $m=2$ or $m=-6$
6. $x=-6$ or $x=-7$
7. $n=\frac{5}{2}$ or $n=-\frac{5}{2}$
8. $p=0$ or $p=\frac{2}{5}$