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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #62 : Completing The Square

Solve for by completing the square.

x^2+18x-20=0

Possible Answers:

$x=6.12\text{ and }x=-0.12$

$x=3.39\text{ and }x=-18.39$

$x=1.05\text{ and }x=-19.05$

$x=9.21\text{ and }x=1.21$

Correct answer:

$x=1.05\text{ and }x=-19.05$

Explanation:

x^2+18x-20=0

Start by adding to both sides so that the terms with the are together on the left side of the equation.

x^2+18x=20

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

x^2+18x+81=20+81

Rewrite the left side of the equation in the squared form.

(x+9)^2=101

Take the square root of both sides.

$x+9=\pm \sqrt{101}$

Now solve for .

$x=-9+\sqrt{101}=1.05$

$x=-9-\sqrt{101}=-19.05$

Round to two places after the decimal.

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Example Question #63 : Completing The Square

Solve for by completing the square.

8x^2-2x-1=0

Possible Answers:

$x=\frac{1}{8}\text{ and }x=-\frac{1}{2}$

$x=\frac{5}{8}\text{ and }x=-\frac{7}{8}$

$x=\frac{1}{2}\text{ and }x=-\frac{1}{4}$

$x=\frac{1}{4}\text{ and }x=-\frac{3}{8}$

Correct answer:

$x=\frac{1}{2}\text{ and }x=-\frac{1}{4}$

Explanation:

8x^2-2x-1=0

Start by adding to both sides so that the terms with the are together on the left side of the equation.

8x^2-2x=1

Next, divide everything by the coefficient of the x^2 term.

$x^2-\frac{1}{4}x=\frac{1}{8}$

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

$x^2-\frac{1}{4}x+\frac{1}{64}=\frac{1}{8}+\frac{1}{64}$

Rewrite the left side of the equation in the squared form.

$(x-\frac{1}{8})^2=\frac{9}{64}$

Take the square root of both sides.

$x-\frac{1}{8}=\pm \sqrt{\frac{9}{64}}$

Now solve for .

$x=\frac{1}{8}+\frac{3}{8}=\frac{1}{2}$

$x=\frac{1}{8}-\frac{3}{8}=-\frac{1}{4}$

Round to two places after the decimal.

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Example Question #66 : Completing The Square

Solve for by completing the square.

x^2+16x+50=0

Possible Answers:

$x=-1.51\text{ and }x=-10.49$

$x=-4.26\text{ and }x=-11.74$

$x=-6.68\text{ and }x=-10.32$

$x=-3.18\text{ and }x=-8.82$

Correct answer:

$x=-4.26\text{ and }x=-11.74$

Explanation:

x^2+16x+50=0

Start by subtracting from both sides so that the terms with the are together on the left side of the equation.

x^2+16x=-50

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

x^2+16x+64=-50+64

Rewrite the left side of the equation in the squared form.

(x+8)^2=14

Take the square root of both sides.

$x+8=\pm \sqrt{14}$

Now solve for .

$x=-8+\sqrt{14}=-4.26$

$x=-8+\sqrt{14}=-11.74$

Round to two places after the decimal.

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Example Question #64 : Completing The Square

Solve for by completing the square.

x^2+20x+50=0

Possible Answers:

$x=-3.07\text{ and }x=-18.93$

$x=-2.93\text{ and }x=-17.07$

$x=-4.73\text{ and }x=-20.27$

$x=-6.27\text{ and }x=-18.73$

Correct answer:

$x=-2.93\text{ and }x=-17.07$

Explanation:

x^2+20x+50=0

Start by subtracting from both sides so that the terms with the are together on the left side of the equation.

x^2+20x=-50

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

x^2+20x+100=-50+100

Rewrite the left side of the equation in the squared form.

(x+10)^2=50

Take the square root of both sides.

$x+10=\pm \sqrt{50}$

Now solve for .

$x=-10+\sqrt{50}=-2.93$

$x=-10+\sqrt{50}=-17.07$

Round to two places after the decimal.

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Example Question #131 : Solving Quadratic Equations

Solve for by completing the square.

3x^2+4x-6=0

Possible Answers:

$x=0.90\text{ and }x=-2.23$

$x=2.22\text{ and }x=-8.09$

$x=-9.37\text{ and }x=5.08$

$x=6.31\text{ and }x=10.81$

Correct answer:

$x=0.90\text{ and }x=-2.23$

Explanation:

3x^2+4x-6=0

Start by adding to both sides so that the terms with the are together on the left side of the equation.

3x^2+4x=6

Next, divide everything by the coefficient of the x^2 term.

$x^2+\frac{4}{3}x=2$

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

$x^2+\frac{4}{3}x+\frac{4}{9}=2+\frac{4}{9}$

Rewrite the left side of the equation in the squared form.

$(x+\frac{2}{3})^2=\frac{22}{9}$

Take the square root of both sides.

$x+\frac{2}{3}=\pm \sqrt{\frac{22}{9}}$

Now solve for .

$x=-\frac{2}{3}+\sqrt{\frac{22}{9}}=0.90$

$x=-\frac{2}{3}-\sqrt{\frac{22}{9}}=-2.23$

Round to two places after the decimal.

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Example Question #65 : Completing The Square

Solve for by completing the square.

10x^2-5x-6=0

Possible Answers:

$x=2.95\text{ and }x=1.05$

$x=27.66\text{ and }x=-2.66$

$x=1.06\text{ and }x=-0.56$

$x=3.51\text{ and }x=0.51$

Correct answer:

$x=1.06\text{ and }x=-0.56$

Explanation:

10x^2-5x-6=0

Start by adding to both sides so that the terms with the are together on the left side of the equation.

10x^2-5x=6

Next, divide everything by the coefficient of the x^2 term.

$x^2-\frac{1}{2}x=\frac{3}{5}$

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

$x^2-\frac{1}{2}x+\frac{1}{16}=\frac{3}{5}+\frac{1}{16}$

Rewrite the left side of the equation in the squared form.

$(x-\frac{1}{4})^2=\frac{53}{80}$

Take the square root of both sides.

$x-\frac{1}{4}=\pm \sqrt{\frac{53}{80}}$

Now solve for .

$x=\frac{1}{4}+\sqrt{\frac{53}{80}}=1.06$

$x=\frac{1}{4}-\sqrt{\frac{53}{80}}=-0.56$

Round to two places after the decimal.

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Example Question #131 : Solving Quadratic Equations

Which number completes the following square equation:

$x^{2}+6x+$ ___

Possible Answers:

Correct answer:

Explanation:

In order to complete the square, you half the middle number and square it.

$x^{2}+6x+$ __________

$\downarrow$ $\uparrow$

$\left(\frac{6}{2}\right)^2=(3)^2=9$

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Example Question #1591 : Algebra Ii

Solve the equation using the quadratic formula:

$\small 0=2x^2+x-7$

Possible Answers:

$\small x=-7\ or\ 8$

$\small x=7\ or\ 8$

$\small x=\frac{-1\pm \sqrt{55}}{4}$

$\small x=\frac{-1\pm \sqrt{57}}{4}$

Correct answer:

$\small x=\frac{-1\pm \sqrt{57}}{4}$

Explanation:

The quadratic formula:

$\small x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\small a=2; b=-1; c=7$

$\small x=\frac{-1\pm \sqrt{1^2-(4)(2)(-7)}}{(2)(2)}=\frac{-1\pm \sqrt{1+56}}{4}=\frac{-1\pm \sqrt{57}}{4}$

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Example Question #241 : Functions And Lines

The product of two consective positive odd integers is 143. Find both integers.

Possible Answers:

$13\ and\ 15$

$-11\ and\ -13$

$11\ and\ 13$

$9\ and\ 11$

$15\ and\ 17$

Correct answer:

$11\ and\ 13$

Explanation:

If is one odd number, then the next odd number is n+2 . If their product is 143, then the following equation is true.

n(n+2) =143

Distribute into the parenthesis.

n^2+2n=143

Subtract 143 from both sides.

$n^{2} + 2n - 143 = 0$

This can be solved by factoring, or by the quadratic equation. We will use the latter.

$n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$n=\frac{-2 \pm \sqrt{4-(4(1)(-143)}}{2} = \frac{-2 \pm \sqrt{576}}{2}=\frac{-2 \pm 24}{2}$

$n=\frac{22}{2}\ or\ n=\frac{-26}{2}$

$n=11\ or\ n=-13$

We are told that both integers are positive, so n=11 .

The other integer is n+2 .

11+2=13

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Example Question #92 : Algebraic Functions

Solve for :

$x^{2}+12x+35=0$

Possible Answers:

x=-5,-9

x=7,9

x=-5,-7

x=3,15

The solution is undefined.

Correct answer:

x=-5,-7

Explanation:

To factor this equation, first find two numbers that multiply to 35 and sum to 12. These numbers are 5 and 7. Split up 12x using these two coefficients:

x^2 + 7x+5x+35=0

x(x+7)+5(x+7)=0

(x+5)(x+7)=0

x=-5,-7

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #62 : Completing The Square

Example Question #63 : Completing The Square

Example Question #66 : Completing The Square

Example Question #64 : Completing The Square

Example Question #131 : Solving Quadratic Equations

Example Question #65 : Completing The Square

Example Question #131 : Solving Quadratic Equations

Example Question #1591 : Algebra Ii

Example Question #241 : Functions And Lines

Example Question #92 : Algebraic Functions

All Algebra II Resources

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