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High School Math : Intermediate Single-Variable Algebra

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Example Questions

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Example Question #1 : Completing The Square

Complete the square:

2n^2+n-21=0

Possible Answers:

$n=-\frac{7}{2},3$

$n=-\frac{5}{2},3$

$n=-\frac{9}{2},3$

$n=-\frac{7}{2},4$

$n=-\frac{7}{2},2$

Correct answer:

$n=-\frac{7}{2},3$

Explanation:

Begin by dividing the equation by and adding 10.5 to each side:

2n^2+n-21=0

$n^2+\frac{1}{2}n = \frac{21}{2}$

Square the value in front of the and add to each side:

$n^2+\frac{1}{2}n + \frac{1}{16} = \frac{21}{2} + \frac{1}{16}$

Factor the left side of the equation:

$(n+\frac{1}{4})^2 = \frac{169}{16}$

Take the square root of both sides and simplify:

$n+\frac{1}{4} = \frac{\pm 13}{4}$

$n = \frac{-1 \pm 13}{4}$

$n=-\frac{7}{2},3$

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Example Question #5 : Completing The Square

Complete the square:

3x^2+4x+2=0

Possible Answers:

$x=\frac{-3 \pm i\sqrt{2}}{3}$

$x=\frac{-4 \pm i\sqrt{2}}{3}$

$x=\frac{-2 \pm i\sqrt{2}}{4}$

$x=\frac{-2 \pm i\sqrt{2}}{3}$

$x=\frac{-2 \pm i\sqrt{3}}{3}$

Correct answer:

$x=\frac{-2 \pm i\sqrt{2}}{3}$

Explanation:

Begin by dividing the equation by and subtracting $\frac{2}{3}$ from each side:

3x^2+4x+2=0

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

Square the value in front of the and add to each side:

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

Factor the left side of the equation:

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

Take the square root of both sides and simplify:

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

$x = \frac{-2 \pm i\sqrt{2}}{3}$

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Example Question #91 : Intermediate Single Variable Algebra

Solve using the quadratic formula:

2x^2+2x+3=0

Possible Answers:

$x= \frac{-2 \pm i\sqrt{5}}{2}$

$x= \frac{-1 \pm i\sqrt{5}}{2}$

$x= \frac{-1 \pm i\sqrt{5}}{3}$

$x= \frac{-3 \pm i\sqrt{5}}{2}$

$x= \frac{-2 \pm i\sqrt{5}}{3}$

Correct answer:

$x= \frac{-1 \pm i\sqrt{5}}{2}$

Explanation:

Use the quadratic formula to solve:

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

a = 2

b = 2

c = 3

$x = \frac{-(2) \pm \sqrt{(2)^2-4(2)(3)}}{2(2)}$

$x = \frac{-2 \pm \sqrt{-20}}{4}$

$x = \frac{-2 \pm 2i\sqrt{5}}{4}$

$x = \frac{-1 \pm i\sqrt{5}}{2}$

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Example Question #51 : Quadratic Equations And Inequalities

Solve using the quadratic formula:

2a^2-5a+4=0

Possible Answers:

$a=\frac{5 \pm i\sqrt{7}}{4}$

$a=\frac{7 \pm i\sqrt{7}}{4}$

$a=\frac{4 \pm i\sqrt{7}}{4}$

$a=\frac{6 \pm i\sqrt{7}}{4}$

$a=\frac{3 \pm i\sqrt{7}}{4}$

Correct answer:

$a=\frac{5 \pm i\sqrt{7}}{4}$

Explanation:

Use the quadratic formula to solve:

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

a = 2

b = -5

c = 4

$a = \frac{-(-5) \pm \sqrt{(-5)^2-4(2)(4)}}{2(2)}$

$a = \frac{5 \pm \sqrt{-7}}{4}$

$a = \frac{5 \pm i\sqrt{7}}{4}$

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Example Question #3 : Using The Quadratic Formula

Solve using the quadratic formula:

ax^2+bx+3b=0

Possible Answers:

$x=\frac{-b \pm \sqrt{b^2-8ab}}{2a}$

$x=\frac{-b \pm \sqrt{b^2-12ab}}{a}$

$x=\frac{b \pm \sqrt{b^2-12ab}}{a}$

$x=\frac{b \pm \sqrt{b^2-12ab}}{2a}$

$x=\frac{-b \pm \sqrt{b^2-12ab}}{2a}$

Correct answer:

$x=\frac{-b \pm \sqrt{b^2-12ab}}{2a}$

Explanation:

Use the quadratic formula to solve:

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

a = a

b = b

c = 3b

$x = \frac{-(b) \pm \sqrt{(b)^2-4(a)(3b)}}{2(a)}$

$x = \frac{-b \pm \sqrt{b^2-12ab}}{2a}$

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Example Question #4 : Using The Quadratic Formula

Solve using the quadratric formula:

bx^2+acx+c=0

Possible Answers:

$x=\frac{ac\sqrt{a^2c^2-4bc}}{2b}$

$x=\frac{-ac\sqrt{a^2c^2-2bc}}{2b}$

$x=\frac{-ac\sqrt{a^2c^2-4bc}}{b}$

$x=\frac{ac\sqrt{a^2c^2-4bc}}{b}$

$x=\frac{-ac\sqrt{a^2c^2-4bc}}{2b}$

Correct answer:

$x=\frac{-ac\sqrt{a^2c^2-4bc}}{2b}$

Explanation:

Use the quadratic formula to solve:

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

a = b

b = ac

c = c

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

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

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Example Question #461 : 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 3.5 and 4 seconds

Between 2.5 and 3 seconds

Between 4 and 4.5 seconds

Between 3 and 3.5 seconds

Between 2 and 2.5 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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Example Question #1 : Solving Quadratic Inequalities

Solve the quadratric inequality:

$y\geq -x^2+10x-21$

Possible Answers:

$x\leq4 , x\geq8$

$x\leq3 , x\geq7$

$x\leq1 , x\geq4$

$x\leq2 , x\geq5$

$x\leq5 , x\geq9$

Correct answer:

$x\leq3 , x\geq7$

Explanation:

Factor and solve.

$y\geq -x^2+10x-21$

$y\leq x^2-10x+21$

Since the equation is less than or equal to, you know the inequality will be OR, not AND.

$y\leq (x-7)(x-3)$

$x\leq3$ or $x\geq7$

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Example Question #2 : Solving Quadratic Inequalities

Solve the following quadratic inequality:

$y\leq x^2+4x+3$

Possible Answers:

$x\leq -3, x\geq -2$

$x\leq -2, x\geq -1$

$x\leq -3, x\geq -1$

$x\leq -4, x\geq -1$

$x\leq -4, x\geq -2$

Correct answer:

$x\leq -3, x\geq -1$

Explanation:

Factor and solve. Since the sign is less than or equal to, we know the inequality will be OR, not AND.

$y\leq x^2+4x+3$

$y\leq (x+3)(x+1)$

$x\leq-3$ or $x\geq -1$

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Example Question #3 : Solving Quadratic Inequalities

Solve the following quadratic inequality:

$y\leq -x^2+6x+5$

Possible Answers:

$3-\sqrt{13}\leq x \leq 3+\sqrt{13}$

$3-\sqrt{14}\leq x \leq 3+\sqrt{14}$

$3-\sqrt{10}\leq x \leq 3+\sqrt{10}$

$3-\sqrt{15}\leq x \leq 3+\sqrt{15}$

$3-\sqrt{11}\leq x \leq 3+\sqrt{11}$

Correct answer:

$3-\sqrt{14}\leq x \leq 3+\sqrt{14}$

Explanation:

Use the quadratic formula to solve.

$y\leq -x^2+6x+5$

$y\geq x^2-6x-5$

Since the inequality is greater than or equal to, we know the inequality will be AND, not OR.

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

a=1

b=-6

c=-5

$x = \frac{-(-6) \pm \sqrt{(-6)^2-4(1)(-5)}}{2(1)}$

$x= \frac{6 \pm \sqrt{56}}{2}$

$x= 3 \pm \sqrt{14}$

$3-\sqrt{14}\leq x \leq 3+\sqrt{14}$

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High School Math : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Completing The Square

Example Question #5 : Completing The Square

Example Question #91 : Intermediate Single Variable Algebra

Example Question #51 : Quadratic Equations And Inequalities

Example Question #3 : Using The Quadratic Formula

Example Question #4 : Using The Quadratic Formula

Example Question #461 : Intermediate Single Variable Algebra

Example Question #1 : Solving Quadratic Inequalities

Example Question #2 : Solving Quadratic Inequalities

Example Question #3 : Solving Quadratic Inequalities

All High School Math Resources

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