Question

for questions 3-12, solve by factoring

1. $x^{2}+5x-36=0$
2. $x^{2}+61=1-17x$
3. $3x^{2}+112=2x^{2}+22x$
4. $2x^{2}+7x=42-x$
5. $12x^{2}+17x=5$
6. $25x^{2}=20x-4$

Explanation:

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Problem 3: $x^2+5x-36=0$

Step1: Factor quadratic trinomial

Find two numbers that multiply to $-36$ and add to $5$: $9$ and $-4$.
$(x+9)(x-4)=0$

Step2: Apply zero product property

Set each factor equal to 0.
$x+9=0$ or $x-4=0$

Step3: Solve for $x$

Calculate solutions from each equation.
$x=-9$ or $x=4$

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Problem 4: $x^2+61=1-17x$

Step1: Rearrange to standard form

Move all terms to left side.
$x^2+17x+60=0$

Step2: Factor quadratic trinomial

Find two numbers that multiply to $60$ and add to $17$: $12$ and $5$.
$(x+12)(x+5)=0$

Step3: Apply zero product property

Set each factor equal to 0.
$x+12=0$ or $x+5=0$

Step4: Solve for $x$

Calculate solutions from each equation.
$x=-12$ or $x=-5$

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Problem 5: $3x^2+112=2x^2+22x$

Step1: Rearrange to standard form

Subtract $2x^2+22x$ from both sides.
$x^2-22x+112=0$

Step2: Factor quadratic trinomial

Find two numbers that multiply to $112$ and add to $-22$: $-14$ and $-8$.
$(x-14)(x-8)=0$

Step3: Apply zero product property

Set each factor equal to 0.
$x-14=0$ or $x-8=0$

Step4: Solve for $x$

Calculate solutions from each equation.
$x=14$ or $x=8$

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Problem 6: $2x^2+7x=42-x$

Step1: Rearrange to standard form

Move all terms to left side.
$2x^2+8x-42=0$

Step2: Simplify the equation

Divide all terms by 2.
$x^2+4x-21=0$

Step3: Factor quadratic trinomial

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

Step4: Apply zero product property

Set each factor equal to 0.
$x+7=0$ or $x-3=0$

Step5: Solve for $x$

Calculate solutions from each equation.
$x=-7$ or $x=3$

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Problem 7: $12x^2+17x=5$

Step1: Rearrange to standard form

Move all terms to left side.
$12x^2+17x-5=0$

Step2: Factor quadratic trinomial

Use AC method: $12*(-5)=-60$, find factors $20$ and $-3$ that add to 17, then rewrite and factor.
$(4x+5)(3x-1)=0$

Step3: Apply zero product property

Set each factor equal to 0.
$4x+5=0$ or $3x-1=0$

Step4: Solve for $x$

Calculate solutions from each equation.
$x=-\frac{5}{4}$ or $x=\frac{1}{3}$

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Problem 8: $25x^2=20x-4$

Step1: Rearrange to standard form

Move all terms to left side.
$25x^2-20x+4=0$

Step2: Factor perfect square trinomial

Recognize $(5x-2)^2=25x^2-20x+4$.
$(5x-2)^2=0$

Step3: Apply zero product property

Set factor equal to 0.
$5x-2=0$

Step4: Solve for $x$

Calculate repeated solution.
$x=\frac{2}{5}$

Answer:

1. $x=-9$ or $x=4$
2. $x=-12$ or $x=-5$
3. $x=14$ or $x=8$
4. $x=-7$ or $x=3$
5. $x=-\frac{5}{4}$ or $x=\frac{1}{3}$
6. $x=\frac{2}{5}$ (repeated root)