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Algebra 1 : Quadratic Equations

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Example Questions

Example Question #61 : Quadratic Equations

Factor the following quadratic expression:

$x^{2}+12x + 32$ $\displaystyle x^{2}+12x + 32$

Possible Answers:

$\displaystyle (x+8)(x+4)$

$\displaystyle (x-32)(x-1)$

$\displaystyle (x+2)(x+8)$

$\displaystyle (x+16)(x+2)$

$\displaystyle (x-8)(x+4)$

Correct answer:

$\displaystyle (x+8)(x+4)$

Explanation:

Given the following expression:

$x^{2}+12x + 32$ $\displaystyle x^{2}+12x + 32$

We need to find factors of $\displaystyle 32$ that add up to $\displaystyle 12$.

$\displaystyle 32$ can be broken down into the following factors:

32 , 1 $\displaystyle 32 , 1$

16, 2 $\displaystyle 16, 2$

8,4 $\displaystyle 8,4$

Of these choices, only 8 + 4 $\displaystyle 8 + 4$ adds up to $\displaystyle 12$. Additionally, the coefficient in front of the variable is $\displaystyle 1$, so we do not need to worry about that when finding these values. There are no negatives in the quadratic expression, so the signs in the factored form are all positive. This gives us the final answer of

$\displaystyle (x+8)(x+4)$

You can use the FOIL method to re-expand the expression and check your work!

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Example Question #12 : How To Factor The Quadratic Equation

Solve the following equation by factoring.

$x^{2}+14x+45=0$ $\displaystyle x^{2}+14x+45=0$

Possible Answers:

None of the other answers.

$\displaystyle (-15,-3)$

(9,5) $\displaystyle (9,5)$

(-9,-5) $\displaystyle (-9,-5)$

$\displaystyle (-11,-4)$

Correct answer:

(-9,-5) $\displaystyle (-9,-5)$

Explanation:

To factor a quadratic equation in the form

$ax^{2}+bx+c$ $\displaystyle ax^{2}+bx+c$, where a=1 $\displaystyle a=1$, find two integers that have a sum of $\displaystyle b$ and a product of $\displaystyle c$.

For this equation, that would be 9 and 5.

9+5=14 $\displaystyle 9+5=14$

$9\cdot 5=45$ $\displaystyle 9\cdot 5=45$

Therefore, the solutions to this equation are $\displaystyle -9$ and $\displaystyle -5$.

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Example Question #112 : Systems Of Equations

Solve the following equation by factoring.

$x^{2}+3x=88$ $\displaystyle x^{2}+3x=88$

Possible Answers:

(-11,8) $\displaystyle (-11,8)$

None of the other answers.

(4,-22) $\displaystyle (4,-22)$

(8,-11) $\displaystyle (8,-11)$

$\displaystyle (-44,41)$

Correct answer:

(-11,8) $\displaystyle (-11,8)$

Explanation:

$x^{2}+3x=88$ $\displaystyle x^{2}+3x=88$

Begin by setting the equation equal to 0 by subtracting 88 from both sides.

$x^{2}+3x-88=0$ $\displaystyle x^{2}+3x-88=0$

Now that the equation is in the form $ax^{2}+bx+c=0$ $\displaystyle ax^{2}+bx+c=0$, find two integers that sum to $\displaystyle b$ and have a product of $\displaystyle c$.

For this equation, those integers are $\displaystyle 11$ and $\displaystyle -8$.

11-8=3 $\displaystyle 11-8=3$

$11\cdot -8=-88$ $\displaystyle 11\cdot -8=-88$

Therefore, the solutions to this equation are (-11,8) $\displaystyle (-11,8)$

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Example Question #2361 : Algebra 1

Solve the following equation by factoring.

$n^{2}+17n-30=-102$ $\displaystyle n^{2}+17n-30=-102$

Possible Answers:

(8,9) $\displaystyle (8,9)$

(17,-6) $\displaystyle (17,-6)$

(-8,-9) $\displaystyle (-8,-9)$

(17,6) $\displaystyle (17,6)$

$\displaystyle (-13,-4)$

Correct answer:

(-8,-9) $\displaystyle (-8,-9)$

Explanation:

$n^{2}+17n-30=-102$ $\displaystyle n^{2}+17n-30=-102$

Begin by setting the equation equal to zero by adding 105 to each side.

$n^{2}+17n+72=0$ $\displaystyle n^{2}+17n+72=0$

For an equation in the form $ax^{2}+bx+c=0$ $\displaystyle ax^{2}+bx+c=0$, where a=1 $\displaystyle a=1$, find two integers that have a sum of $\displaystyle b$ and a product of $\displaystyle c$.

For this equation, that would be 8 and 9.

8+9=17 $\displaystyle 8+9=17$

$8\cdot 9=72$ $\displaystyle 8\cdot 9=72$

Therefore, the solutions to this equation are (-8,-9) $\displaystyle (-8,-9)$

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Algebra 1 : Quadratic Equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #61 : Quadratic Equations

Example Question #12 : How To Factor The Quadratic Equation

Example Question #112 : Systems Of Equations

Example Question #2361 : Algebra 1

All Algebra 1 Resources

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