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High School Math : Quadratic Equations and Inequalities

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Example Questions

Example Question #1 : Understanding Quadratic Equations

Solve the following equation using the quadratic form:

x^4-9=0

Possible Answers:

$x=\pm\sqrt{6}, \pm i\sqrt{6}$

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

$x=\pm\sqrt{5}, \pm i\sqrt{5}$

$x=\pm\sqrt{4}, \pm i\sqrt{4}$

$x=\pm\sqrt{2}, \pm i\sqrt{2}$

Correct answer:

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

Explanation:

Factor the equation and solve:

x^4-9=0

(x^2-3)(x^2+3)=0

x^2 = 3

$x = \pm \sqrt{3}$

x^2 = -3

$x = \pm i\sqrt{3}$

Therefore there are four answers:

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

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Example Question #2 : Understanding Quadratic Equations

Solve the following equation using the quadratic form:

$x-4\sqrt{x}-45=0$

Possible Answers:

x=25

x=81

x=36

x=49

x=64

Correct answer:

x=81

Explanation:

Factor the equation and solve:

$x-4\sqrt{x}-45=0$

$(\sqrt{x}-9)(\sqrt{x}+5)=0$

$\sqrt{x}=9$

x=81

$\sqrt{x}=-5$

This has no solutions.

Therefore there is only one answer:

x=81

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Example Question #3 : Understanding Quadratic Equations

Evaluate $(2x+2y)^{2}$

Possible Answers:

$4x^{2}+8xy+4y^{2}$

$\dpi{100} x^{2}+xy+y^{2}$

$\dpi{100} 4x^{2}+4y^{2}$

$\dpi{100} 2x^{2}+4xy+2y^{2}$

$\dpi{100} 4x^{2}+4xy+4y^{2}$

Correct answer:

$4x^{2}+8xy+4y^{2}$

Explanation:

In order to evaluate $\dpi{100} (2x+2y)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.

$\dpi{100} \dpi{100} (2x+2y)^{2}=(2x+2y)(2x+2y)$

Multiply terms by way of FOIL method.

=(2x*2x)+(2x*2y)+(2y*2x)+(2y*2y)

Now multiply and simplify.

$=4x^{2}+4xy+4xy+4y^{2}$

$\rightarrow 4x^{2}+8xy+4y^{2}$

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Example Question #4 : Understanding Quadratic Equations

Evaluate $(x-2)^{2}$

Possible Answers:

$x^{2}-4x+4$

$\dpi{100} x^{2}-2x+4$

$\dpi{100} x^{2}-4$

$\dpi{100} x^{2}+4$

$\dpi{100} 4x^{2}-4x+4$

Correct answer:

$x^{2}-4x+4$

Explanation:

In order to evaluate $\dpi{100} \dpi{100} (x-2)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms. Be sure to pay attention to signs.

$(x-2)^{2}=(x-2)(x-2)$

Multiply terms by way of FOIL method.

=(x*x)+(x*-2)+(-2*x)+(-2*-2)

Now multiply and simplify, paying attention to signs.

$=x^{2}+(-2x)+(-2x)+4$

$\rightarrow x^{2}-4x+4$

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Example Question #5 : Understanding Quadratic Equations

Evaluate (x+2y)*(2x-y)

Possible Answers:

$\dpi{100} 4x^{2}+xy-4y^{2}$

$\dpi{100} x^{2}+2xy-y^{2}$

$\dpi{100} -2x^{2}-3xy+2y^{2}$

$2x^{2}+3xy-2y^{2}$

$\dpi{100} 2x^{2}+4xy-2y^{2}$

Correct answer:

$2x^{2}+3xy-2y^{2}$

Explanation:

In order to evaluate (x+2y)*(2x-y) one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms. Be sure to pay attention to signs.

Multiply terms by way of FOIL method.

=(x*2x)+(x*-y)+(2y*2x)+(2y*-y)

Now multiply and simplify, paying attention to signs.

$=2x^{2}+(-xy)+4xy+(-2y^{2})$

$\rightarrow 2x^{2}+3xy-2y^{2}$

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Example Question #6 : Understanding Quadratic Equations

Evaluate $(x-y)^{2}$

Possible Answers:

$x^{2}+y^{2}$

$\dpi{100} x^{2}-y^{2}$

$\dpi{100} -x^{2}+2xy-y^{2}$

$\dpi{100} 2x^{2}-4xy+2y^{2}$

$x^{2}-2xy+y^{2}$

Correct answer:

$x^{2}-2xy+y^{2}$

Explanation:

In order to evaluate $\dpi{100} \dpi{100} (x-y)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms. Be sure to pay attention to signs.

$(x-y)^{2}=(x-y)(x-y)$

Multiply terms by way of FOIL method.

=(x*x)+(x*-y)+(-y*x)+(-y*-y)

Now multiply and simplify, paying attention to signs.

$=x^{2}+(-xy)+(-xy)+(y^{2})$

$\rightarrow x^{2}-2xy+y^{2}$

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Example Question #7 : Understanding Quadratic Equations

FOIL (x+2)(x-2) .

Possible Answers:

x^2-4

x^2+x-2

x^2+4x+4

x^2-2x+2

x^2+4

Correct answer:

x^2-4

Explanation:

Remember FOIL stands for First Outer Inner Last. That means we can take (x+2)(x-2) and turn it into x^2+2x-2x-4 .

Simplify to get x^2-4 .

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Example Question #8 : Understanding Quadratic Equations

FOIL (x+5)^2 .

Possible Answers:

x^2+25

x^2+10x+25

x^2+25x+25

5x^2

x^2-10x+25

Correct answer:

x^2+10x+25

Explanation:

(x+5)^2 is the same thing as (x+5)(x+5) .

Remember that FOIL stands for First Outer Inner Last.

For this problem, that would be x^2+5x+5x+25 .

Simplify that to x^2+10x+25 .

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Example Question #9 : Understanding Quadratic Equations

FOIL (x+5)(x-1) .

Possible Answers:

x^2-5

x^2+5x+5

x^2-4x-5

x^2+4x-5

x^2+24

Correct answer:

x^2+4x-5

Explanation:

Remember that FOIL stands for First Outer Inner Last.

For this problem that would give us:

(x+5)(x-1)

x^2-x+5x-5

Simplify.

x^2+4x-5

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Example Question #10 : Understanding Quadratic Equations

FOIL (x+2)(x-2) .

Possible Answers:

x^2-4x-4

x^2-4

x^2+4

-x^2+4

x^2-4x

Correct answer:

x^2-4

Explanation:

Remember that FOIL stands for First Outer Inner Last.

For this problem that would give us:

(x+2)(x-2)

=x^2-2x+2x-4

Simplify:

=x^2-4

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High School Math : Quadratic Equations and Inequalities

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Quadratic Equations

Example Question #2 : Understanding Quadratic Equations

Example Question #3 : Understanding Quadratic Equations

Example Question #4 : Understanding Quadratic Equations

Example Question #5 : Understanding Quadratic Equations

Example Question #6 : Understanding Quadratic Equations

Example Question #7 : Understanding Quadratic Equations

Example Question #8 : Understanding Quadratic Equations

Example Question #9 : Understanding Quadratic Equations

Example Question #10 : Understanding Quadratic Equations

All High School Math Resources

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