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High School Math : Finding Roots

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Example Questions

Example Question #1 : Finding Roots

Find the zeros.

$f(t)=t^{2}-4t-21$ $\displaystyle f(t)=t^{2}-4t-21$

Possible Answers:

x=-3,7 $\displaystyle x=-3,7$

x=-7,3 $\displaystyle x=-7,3$

x=7,-7 $\displaystyle x=7,-7$

x=3,-3 $\displaystyle x=3,-3$

x=7,3 $\displaystyle x=7,3$

Correct answer:

x=-3,7 $\displaystyle x=-3,7$

Explanation:

Factor the equation to $\displaystyle (t-7)(t+3)$. Set both equal to zero and you get $\displaystyle 7$ and $\displaystyle -3$. Remember, the zeros of an equation are wherever the function crosses the $\displaystyle x$-axis.

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Example Question #2 : Finding Roots

Find the zeros.

$f(x)=x^{2}-10x$ $\displaystyle f(x)=x^{2}-10x$

Possible Answers:

x=10 $\displaystyle x=10$

x=0 $\displaystyle x=0$

x=-10 $\displaystyle x=-10$

x=0,10 $\displaystyle x=0,10$

x=1,-1 $\displaystyle x=1,-1$

Correct answer:

x=0,10 $\displaystyle x=0,10$

Explanation:

Factor out an $\displaystyle x$ from the equation so that you have x(x-10) $\displaystyle x(x-10)$. Set $\displaystyle x$ and x-10 $\displaystyle x-10$ equal to $\displaystyle 0$. Your roots are $\displaystyle 0$ and $\displaystyle 10$.

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Example Question #3 : Finding Roots

Find the zeros.

$f(x)=\frac{1}{5}(x+9)^{2}(x^{2}-4)$ $\displaystyle f(x)=\frac{1}{5}(x+9)^{2}(x^{2}-4)$

Possible Answers:

x=2,-9 $\displaystyle x=2,-9$

x=-2,2 $\displaystyle x=-2,2$

x=-9 $\displaystyle x=-9$

$\displaystyle x=-3,3,-9$

$\displaystyle x=-9,-2,2$

Correct answer:

$\displaystyle x=-9,-2,2$

Explanation:

Set (x+9)^2 $\displaystyle (x+9)^2$ equal to zero and you get $\displaystyle -9$. Set (x^2-4) $\displaystyle (x^2-4)$ equal to zero as well and you get $\displaystyle 2$ and $\displaystyle -2$ because when you take a square root, your answer will be positive and negative.

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Example Question #4 : Finding Roots

Find the zeros.

$f(x)=27x^{2}-45x-18$ $\displaystyle f(x)=27x^{2}-45x-18$

Possible Answers:

$x=9,-\frac{1}{3},2$ $\displaystyle x=9,-\frac{1}{3},2$

$x=\frac{1}{3},2$ $\displaystyle x=\frac{1}{3},2$

x=-2,-9 $\displaystyle x=-2,-9$

$x=\frac{1}{3}$ $\displaystyle x=\frac{1}{3}$

$x=-\frac{1}{3},2$ $\displaystyle x=-\frac{1}{3},2$

Correct answer:

$x=-\frac{1}{3},2$ $\displaystyle x=-\frac{1}{3},2$

Explanation:

Factor out a $\displaystyle 9$ from the entire equation. After that, you get $\displaystyle 9(3x^2-5x-2)$. Factor the expression to $\displaystyle (3x+1)(x-2)$. Set both of those equal to zero and your answers are $-\frac{1}{3}$ $\displaystyle -\frac{1}{3}$ and $\displaystyle 2$.

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Example Question #5 : Finding Roots

Find the zeros.

$f(x)=25x^{2}-81$ $\displaystyle f(x)=25x^{2}-81$

Possible Answers:

$x=-\frac{9}{5}$ $\displaystyle x=-\frac{9}{5}$

$x=\frac{9}{5}$ $\displaystyle x=\frac{9}{5}$

x=9,-9 $\displaystyle x=9,-9$

$x=\frac{9}{5},-\frac{9}{5}$ $\displaystyle x=\frac{9}{5},-\frac{9}{5}$

x=5 $\displaystyle x=5$

Correct answer:

$x=\frac{9}{5},-\frac{9}{5}$ $\displaystyle x=\frac{9}{5},-\frac{9}{5}$

Explanation:

This expression is the difference of perfect squares. Therefore, it factors to$\displaystyle (5x-9)(5x+9)$. Set both of those equal to zero and your answers are $\frac{9}{5}$ $\displaystyle \frac{9}{5}$ and $-\frac{9}{5}$ $\displaystyle -\frac{9}{5}$.

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Example Question #6 : Finding Roots

Find the zeros.

$\displaystyle f(x)= 36x^2-9x$

Possible Answers:

$x=0,\frac{1}{4}$ $\displaystyle x=0,\frac{1}{4}$

x=8 $\displaystyle x=8$

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

x=0 $\displaystyle x=0$

x=0,4 $\displaystyle x=0,4$

Correct answer:

$x=0,\frac{1}{4}$ $\displaystyle x=0,\frac{1}{4}$

Explanation:

Factor the equation to $\displaystyle 9x(4x-1)$. Set both equal to $\displaystyle 0$ and you get $\displaystyle 0$ and $\frac{1}{4}$ $\displaystyle \frac{1}{4}$.

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Example Question #7 : Finding Roots

Find the zeros.

$f(x)=\frac{1}{4}x^2 +\frac{5}{4}x+\frac{3}{2}$ $\displaystyle f(x)=\frac{1}{4}x^2 +\frac{5}{4}x+\frac{3}{2}$

Possible Answers:

x=3,2 $\displaystyle x=3,2$

x=4,2 $\displaystyle x=4,2$

x=-3,-2 $\displaystyle x=-3,-2$

x=2 $\displaystyle x=2$

x=-3,3 $\displaystyle x=-3,3$

Correct answer:

x=-3,-2 $\displaystyle x=-3,-2$

Explanation:

Factor a $\frac{1}{4}$ $\displaystyle \frac{1}{4}$ out of the quation to get

$\frac{1}{4}\left ( x^{2}+5x+6 \right )$ $\displaystyle \frac{1}{4}\left ( x^{2}+5x+6 \right )$

which can be further factored to

$\frac{1}{4}(x+3)(x+2)$ $\displaystyle \frac{1}{4}(x+3)(x+2)$.

Set the last two expressions equal to zero and you get $\displaystyle -3$ and $\displaystyle -2$.

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Example Question #8 : Finding Roots

Find the zeros.

$\displaystyle f(x)=x(x-6)^2$

Possible Answers:

x=1,-1 $\displaystyle x=1,-1$

x=6 $\displaystyle x=6$

x=0,6 $\displaystyle x=0,6$

x=0 $\displaystyle x=0$

x=-6,6 $\displaystyle x=-6,6$

Correct answer:

x=0,6 $\displaystyle x=0,6$

Explanation:

Set each expression equal to zero and you get 0 and 6.

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Example Question #9 : Finding Roots

Find the zeros.

$\displaystyle f(x)=(x-2)(x+4)^3$

Possible Answers:

x=2 $\displaystyle x=2$

x=2,-4 $\displaystyle x=2,-4$

$\displaystyle x=0,2,-2$

x=4 $\displaystyle x=4$

x=-2,4 $\displaystyle x=-2,4$

Correct answer:

x=2,-4 $\displaystyle x=2,-4$

Explanation:

Set both expressions equal to $\displaystyle 0$. The first factor yields $\displaystyle 2$. The second factor gives you $\displaystyle -4$.

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Example Question #10 : Finding Roots

Find the zeros.

$\displaystyle f(x)=(x+5)(x-8)^2$

Possible Answers:

x=-8 $\displaystyle x=-8$

x=-5,8 $\displaystyle x=-5,8$

x=5,8 $\displaystyle x=5,8$

x=-5 $\displaystyle x=-5$

x=0 $\displaystyle x=0$

Correct answer:

x=-5,8 $\displaystyle x=-5,8$

Explanation:

Set both expressions to $\displaystyle 0$ and you get $\displaystyle -5$ and $\displaystyle 8$.

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High School Math : Finding Roots

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Finding Roots

Example Question #2 : Finding Roots

Example Question #3 : Finding Roots

Example Question #4 : Finding Roots

Example Question #5 : Finding Roots

Example Question #6 : Finding Roots

Example Question #7 : Finding Roots

Example Question #8 : Finding Roots

Example Question #9 : Finding Roots

Example Question #10 : Finding Roots

All High School Math Resources

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