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Algebra II : Quadratic Equations and Inequalities

Study concepts, example questions & explanations for Algebra II

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Example Questions

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 #1 : How To Find Consecutive Integers

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

Possible Answers:

$13\ and\ 15$

$15\ and\ 17$

$11\ and\ 13$

$-11\ and\ -13$

$9\ and\ 11$

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 #291 : Quadratic Equations And Inequalities

Solve for :

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

Possible Answers:

x=3,15

x=-5,-9

The solution is undefined.

x=7,9

x=-5,-7

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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Example Question #94 : Intermediate Single Variable Algebra

A baseball that is thrown in the air follows a trajectory of h(t)= -4t^2 +12t+6 , where h(t) is the height of the ball in feet and is the time elapsed in seconds. How long does the ball stay in the air before it hits the ground?

Possible Answers:

Between 2.5 and 3 seconds

Between 2 and 2.5 seconds

Between 3.5 and 4 seconds

Between 3 and 3.5 seconds

Between 4 and 4.5 seconds

Correct answer:

Between 3 and 3.5 seconds

Explanation:

To solve this, we look at the equation h(t)=0 .

Setting the equation equal to 0 we get 0= -4t^2 +12t+6 .

Once in this form, we can use the Quadratic Formula to solve for .

The quadratic formula says that if 0= -ax^2 +bx+c , then

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

Plugging in our values:

$t= \frac{-12\pm\sqrt{(12^2-4(-4)(6)}}{2(-4)}=\frac{-12\pm \sqrt{240}}{-8}= \frac{12\pm15.5}{8}$

Therefore t=3.4375 or -.4375 and since we are looking only for positive values (because we can't have negative time), 3.4375 seconds is our answer.

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Algebra II : Quadratic Equations and Inequalities

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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 #1 : How To Find Consecutive Integers

Example Question #291 : Quadratic Equations And Inequalities

Example Question #94 : Intermediate Single Variable Algebra

All Algebra II Resources

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