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Algebra II : Intermediate Single-Variable Algebra

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 #281 : Quadratic Equations And Inequalities

Solve for by completing the square.

3x^2-5x-10=0

Possible Answers:

$x=3.69\text{ and }x=-2.31$

$x=7.08\text{ and }x=-3.92$

$x=4.09\text{ and }x=1.91$

$x=2.84\text{ and }x=-1.17$

Correct answer:

$x=2.84\text{ and }x=-1.17$

Explanation:

3x^2-5x-10=0

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

3x^2-5x=10

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

$x^2-\frac{5}{3}x=\frac{10}{3}$

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.

$x^2-\frac{5}{3}x+\frac{25}{36}=\frac{10}{3}+\frac{25}{36}$

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

$(x-\frac{5}{6})^2=\frac{145}{36}$

Take the square root of both sides.

$x-\frac{5}{6}=\pm \sqrt{\frac{145}{36}}$

Now solve for .

$x=\frac{5}{6}+\sqrt{\frac{145}{36}}=2.84$

$x=\frac{5}{6}-\sqrt{\frac{145}{36}}=-1.17$

Round to two places after the decimal.

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Example Question #63 : Completing The Square

Solve for by completing the square.

x^2-10x-12=0

Possible Answers:

$x=20.75\text{ and }x=8.75$

$x=12.36\text{ and }x=2.36$

$x=11.08\text{ and }x=-1.08$

$x=-3.46\text{ and }x=-16.46$

Correct answer:

$x=11.08\text{ and }x=-1.08$

Explanation:

x^2-10x-12=0

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

x^2-10x=12

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^2-10x+25=12+25

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

(x-5)^2=37

Take the square root of both sides.

$x-5=\pm \sqrt{37}$

Now solve for .

$x=5+\sqrt{37}=11.08$

$x=5-\sqrt{37}=-1.08$

Round to two places after the decimal.

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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.

Report an Error

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.

Report an Error

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.

Report an Error

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.

Report an Error

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 #1 : Quadratic Formula

Solve the equation using the quadratic formula:

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

Possible Answers:

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

$\small x=7\ or\ 8$

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

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

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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Algebra II : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #281 : Quadratic Equations And Inequalities

Example Question #63 : Completing The Square

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 #1 : Quadratic Formula

All Algebra II Resources

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