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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 #34 : Finding Roots

Find the roots of the given quadratic.

x^2+10x-24

Possible Answers:

(x+12)(x-2)

(x-12)(x+2)

(x+14)(x-2)

(x-12)(x-2)

(x+12)(x+2)

Correct answer:

(x+12)(x-2)

Explanation:

The roots of a quadratic can be found by factoring. Factoring the solution will give you an expression that when multiplying will result in the initial quadratic.

The roots for this expression are (x+12)(x-2) . Similarly the roots of a quadratic can be found using the quadratic quadratic formula where the parent function of a quadratic is represented as ax^2+bx+c and the quadratic formula is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

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Example Question #33 : Solving Quadratic Equations

What are the roots of the following graph?

Graph for questions

Possible Answers:

x=-1

x=5

y=9

None of the above

y=-1

y=5

x=2

Correct answer:

x=-1

x=5

Explanation:

Roots when looking at a graph are simply the x-values of where the function crosses the x-axis.

Which, when looking at this graph, it is clearly

x=-1 and x=5

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Example Question #34 : Solving Quadratic Equations

Solve the equation.

3d^2+81=0

Possible Answers:

$d=\pm\3$

$d=\pm\3i \sqrt{3}$

$d=\pm\sqrt{3}$

$d=\pm\3i$

Correct answer:

$d=\pm\3i \sqrt{3}$

Explanation:

To solve this equation perform the oppisite operation to isolate the variable.

3d^2+81=0

3d^2=-81

d^2=-27

$d=\pm \sqrt{-27}$

Recall that the square root of a negative number results in an imaginary number.

i^2=-1

Therefore,

$d=\pm i \sqrt{27}$

$d=\pm\3i \sqrt{3}$ .

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Example Question #33 : Finding Roots

Solve the equation.

6x^2=-126

Possible Answers:

$x=\pm{i\sqrt{21}}$

$x=\pm{\sqrt{21}}$

$x=\pm{\sqrt{-21}}$

$x=\pm{7\sqrt{-3}}$

Correct answer:

$x=\pm{i\sqrt{21}}$

Explanation:

To solve this equation perform the oppisite operation to isolate the variable.

6x^2=-126

x^2=-21

Recall that the square root of a negative number is an imaginary number.

i^2=-1

Therefore,

$x=\pm{\sqrt{-21}}$

$x=\pm{i\sqrt{21}}$ .

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Example Question #34 : Solving Quadratic Equations

What are the x-intercepts for f(x)=x^2-3x-4?

Possible Answers:

x=1,-4

x=-1,4

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

x=-1,1

x=4

Correct answer:

x=-1,4

Explanation:

The x-intercepts of a quadratic equation are also the solutions. To find them, factor the quadratic equation. After some trial and error, it can be factored to: (x-4)(x+1) . Set those expressions equal to to get you x-intercepts. Your answers are: x=-1,4 .

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Example Question #35 : Solving Quadratic Equations

Find all solutions to the following quadratic equation:

x^2-225=0

Possible Answers:

x=5,10

x=-15,0

$x=\pm15$

x=15,0

Correct answer:

$x=\pm15$

Explanation:

Find all solutions to the following quadratic equation:

x^2-225=0

We can solve the following equation by first bringing the -225 to the other side:

x^2=225

Next, take the square root of both sides.

$x=\sqrt{225}$

Now, you may be tempted to write your answer as just

$x=\sqrt{225}=15$

But, we need to remeber the following is also true

(-15)^2=225

So our answer choice must include both positive and negative 15

$x=\pm15$

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Example Question #191 : Quadratic Equations And Inequalities

What are the solutions of x^2-5x+4=0?

Possible Answers:

x=-1,4

x=4

x=1,-4

x=1

x=1,4

Correct answer:

x=1,4

Explanation:

To find the roots, or solutions, of the equation, factor the quadratic. It factors to (x-4)(x-1) . Then, set each expression equal to 0 to get your roots of 1 and 4.

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Example Question #192 : Quadratic Equations And Inequalities

What is the x-intercept of f(x)=x^2-1 ?

Possible Answers:

x=0

x=1

x=1,-1

x=-1

Correct answer:

x=1,-1

Explanation:

An x-intercept is the same thing as a root or solution. Therefore, we can set the function equal to 0 and solve for x. $x^2-1=0; x^2=1; x=\pm 1$ .

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Example Question #193 : Quadratic Equations And Inequalities

Find the roots of the equation shown below:

4x^3-4x^2-120x

Possible Answers:

6, -5

-6, 5

-6, 0, 5

0, 5, 6

-5, 0, 6

Correct answer:

-5, 0, 6

Explanation:

The first step in finding the root is to factor the polynomial. The common factor in each term is , so extract that from the equation first. 4x(x^2-x-30) . Continue to find the numbers that will multiply to , and add to . If there is no number in front of the variable, it means there is just one of them!

The factors of 30=(1,2,3,5,6,10,15,30) . One of the terms will need to be negative, and in this case it will be the larger number in order to equal . The numbers that work are 5,6 . The factored polynomial would be 4x(x-6)(x+5) . There are three roots for this problem. The first will be 0, because if the is multiplied by , then it will equal 0. The other numbers to fulfill the equation would be 6, -5 . Therefore, the roots are (-5,0,6) .

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Example Question #193 : Quadratic Equations And Inequalities

What are the roots of the following equation?

(x+3)(x)

Possible Answers:

x=3, x=0

x=-3

x=-3, x=0

x=3

x=3, x=-3

Correct answer:

x=-3, x=0

Explanation:

Roots are the values of "x" where your function equals 0. You need to set both equations equal to 0 and then solve for "x." Upon doing so you get:

x=-3, x=0

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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 #34 : Finding Roots

Example Question #33 : Solving Quadratic Equations

Example Question #34 : Solving Quadratic Equations

Example Question #33 : Finding Roots

Example Question #34 : Solving Quadratic Equations

Example Question #35 : Solving Quadratic Equations

Example Question #191 : Quadratic Equations And Inequalities

Example Question #192 : Quadratic Equations And Inequalities

Example Question #193 : Quadratic Equations And Inequalities

Example Question #193 : Quadratic Equations And Inequalities

All Algebra II Resources

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