High School Math : Understanding Quadratic Equations

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Understanding Quadratic Equations

Solve the following equation using the quadratic form:

\(\displaystyle x^4-9=0\)

Possible Answers:

\(\displaystyle x=\pm\sqrt{4}, \pm i\sqrt{4}\)

\(\displaystyle x=\pm\sqrt{6}, \pm i\sqrt{6}\)

\(\displaystyle x=\pm\sqrt{2}, \pm i\sqrt{2}\)

\(\displaystyle x=\pm\sqrt{5}, \pm i\sqrt{5}\)

\(\displaystyle x=\pm\sqrt{3}, \pm i\sqrt{3}\)

Correct answer:

\(\displaystyle x=\pm\sqrt{3}, \pm i\sqrt{3}\)

Explanation:

Factor the equation and solve:

\(\displaystyle x^4-9=0\)

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

\(\displaystyle x^2 = 3\)

\(\displaystyle x = \pm \sqrt{3}\)

 or

\(\displaystyle x^2 = -3\)

\(\displaystyle x = \pm i\sqrt{3}\)

Therefore there are four answers:

\(\displaystyle x=\pm\sqrt{3}, \pm i\sqrt{3}\)

Example Question #2 : Understanding Quadratic Equations

Solve the following equation using the quadratic form:

\(\displaystyle x-4\sqrt{x}-45=0\)

Possible Answers:

\(\displaystyle x=36\)

\(\displaystyle x=49\)

\(\displaystyle x=81\)

\(\displaystyle x=25\)

\(\displaystyle x=64\)

Correct answer:

\(\displaystyle x=81\)

Explanation:

Factor the equation and solve:

\(\displaystyle x-4\sqrt{x}-45=0\)

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

\(\displaystyle \sqrt{x}=9\)

\(\displaystyle x=81\)

or

\(\displaystyle \sqrt{x}=-5\)

This has no solutions.

Therefore there is only one answer:

\(\displaystyle x=81\)

Example Question #1 : Understanding Quadratic Equations

Evaluate \(\displaystyle (2x+2y)^{2}\)

Possible Answers:

\(\displaystyle \dpi{100} 4x^{2}+4xy+4y^{2}\)

\(\displaystyle 4x^{2}+8xy+4y^{2}\)

\(\displaystyle \dpi{100} x^{2}+xy+y^{2}\)

\(\displaystyle \dpi{100} 4x^{2}+4y^{2}\)

\(\displaystyle \dpi{100} 2x^{2}+4xy+2y^{2}\)

Correct answer:

\(\displaystyle 4x^{2}+8xy+4y^{2}\)

Explanation:

In order to evaluate \(\displaystyle \dpi{100} (2x+2y)^{2}\) 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.

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

Multiply terms by way of FOIL method.

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

Now multiply and simplify.

\(\displaystyle =4x^{2}+4xy+4xy+4y^{2}\)

\(\displaystyle \rightarrow 4x^{2}+8xy+4y^{2}\)

Example Question #2 : Understanding Quadratic Equations

Evaluate \(\displaystyle (x-2)^{2}\)

Possible Answers:

\(\displaystyle \dpi{100} x^{2}+4\)

\(\displaystyle x^{2}-4x+4\)

\(\displaystyle \dpi{100} 4x^{2}-4x+4\)

\(\displaystyle \dpi{100} x^{2}-4\)

\(\displaystyle \dpi{100} x^{2}-2x+4\)

Correct answer:

\(\displaystyle x^{2}-4x+4\)

Explanation:

In order to evaluate \(\displaystyle \dpi{100} \dpi{100} (x-2)^{2}\) 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.

\(\displaystyle (x-2)^{2}=(x-2)(x-2)\)

Multiply terms by way of FOIL method.

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

Now multiply and simplify, paying attention to signs.

\(\displaystyle =x^{2}+(-2x)+(-2x)+4\)

\(\displaystyle \rightarrow x^{2}-4x+4\)

Example Question #1 : Quadratic Equations And Inequalities

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

Possible Answers:

\(\displaystyle 2x^{2}+3xy-2y^{2}\)

\(\displaystyle \dpi{100} x^{2}+2xy-y^{2}\)

\(\displaystyle \dpi{100} 2x^{2}+4xy-2y^{2}\)

\(\displaystyle \dpi{100} -2x^{2}-3xy+2y^{2}\)

\(\displaystyle \dpi{100} 4x^{2}+xy-4y^{2}\)

Correct answer:

\(\displaystyle 2x^{2}+3xy-2y^{2}\)

Explanation:

In order to evaluate \(\displaystyle (x+2y)*(2x-y)\) 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.

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

Now multiply and simplify, paying attention to signs.

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

\(\displaystyle \rightarrow 2x^{2}+3xy-2y^{2}\)

Example Question #3 : Understanding Quadratic Equations

Evaluate \(\displaystyle (x-y)^{2}\)

Possible Answers:

\(\displaystyle x^{2}-2xy+y^{2}\)

\(\displaystyle \dpi{100} -x^{2}+2xy-y^{2}\)

\(\displaystyle \dpi{100} x^{2}-y^{2}\)

\(\displaystyle x^{2}+y^{2}\)

\(\displaystyle \dpi{100} 2x^{2}-4xy+2y^{2}\)

Correct answer:

\(\displaystyle x^{2}-2xy+y^{2}\)

Explanation:

In order to evaluate \(\displaystyle \dpi{100} \dpi{100} (x-y)^{2}\) 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.

\(\displaystyle (x-y)^{2}=(x-y)(x-y)\)

Multiply terms by way of FOIL method.

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

Now multiply and simplify, paying attention to signs.

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

\(\displaystyle \rightarrow x^{2}-2xy+y^{2}\)

Example Question #1 : Quadratic Equations And Inequalities

FOIL \(\displaystyle (x+2)(x-2)\).

Possible Answers:

\(\displaystyle x^2+x-2\)

\(\displaystyle x^2+4x+4\)

\(\displaystyle x^2+4\)

\(\displaystyle x^2-2x+2\)

\(\displaystyle x^2-4\)

Correct answer:

\(\displaystyle x^2-4\)

Explanation:

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

Simplify to get \(\displaystyle x^2-4\).

Example Question #4 : Understanding Quadratic Equations

FOIL \(\displaystyle (x+5)^2\).

Possible Answers:

\(\displaystyle 5x^2\)

\(\displaystyle x^2-10x+25\)

\(\displaystyle x^2+25\)

\(\displaystyle x^2+10x+25\)

\(\displaystyle x^2+25x+25\)

Correct answer:

\(\displaystyle x^2+10x+25\)

Explanation:

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

Remember that FOIL stands for First Outer Inner Last.

For this problem, that would be \(\displaystyle x^2+5x+5x+25\)

Simplify that to \(\displaystyle x^2+10x+25\).

Example Question #3 : Understanding Quadratic Equations

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

Possible Answers:

\(\displaystyle x^2+4x-5\)

\(\displaystyle x^2-4x-5\)

\(\displaystyle x^2-5\)

\(\displaystyle x^2+24\)

\(\displaystyle x^2+5x+5\)

Correct answer:

\(\displaystyle x^2+4x-5\)

Explanation:

Remember that FOIL stands for First Outer Inner Last.

For this problem that would give us:

\(\displaystyle (x+5)(x-1)\)

\(\displaystyle x^2-x+5x-5\)

Simplify.

\(\displaystyle x^2+4x-5\)

Example Question #5 : Understanding Quadratic Equations

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

Possible Answers:

\(\displaystyle -x^2+4\)

\(\displaystyle x^2-4x\)

\(\displaystyle x^2-4\)

\(\displaystyle x^2+4\)

\(\displaystyle x^2-4x-4\)

Correct answer:

\(\displaystyle x^2-4\)

Explanation:

Remember that FOIL stands for First Outer Inner Last.

For this problem that would give us:

\(\displaystyle (x+2)(x-2)\)

\(\displaystyle =x^2-2x+2x-4\)

Simplify:

\(\displaystyle =x^2-4\)

 

 

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