High School Math : Solving Quadratic Equations

Study concepts, example questions & explanations for High School Math

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Solving Quadratic Equations

Find the zeros.

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

Possible Answers:

\(\displaystyle x=7,3\)

\(\displaystyle x=-7,3\)

\(\displaystyle x=3,-3\)

\(\displaystyle x=-3,7\)

\(\displaystyle x=7,-7\)

Correct answer:

\(\displaystyle x=-3,7\)

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.

Example Question #2 : Solving Quadratic Equations

Find the zeros.

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

Possible Answers:

\(\displaystyle x=0\)

\(\displaystyle x=1,-1\)

\(\displaystyle x=10\)

\(\displaystyle x=-10\)

\(\displaystyle x=0,10\)

Correct answer:

\(\displaystyle x=0,10\)

Explanation:

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

Example Question #3 : Solving Quadratic Equations

Find the zeros.

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

Possible Answers:

\(\displaystyle x=-9,-2,2\)

\(\displaystyle x=2,-9\)

\(\displaystyle x=-9\)

\(\displaystyle x=-2,2\)

\(\displaystyle x=-3,3,-9\)

Correct answer:

\(\displaystyle x=-9,-2,2\)

Explanation:

Set \(\displaystyle (x+9)^2\) equal to zero and you get \(\displaystyle -9\). Set \(\displaystyle (x^2-4)\) 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.

Example Question #4 : Solving Quadratic Equations

Find the zeros.

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

Possible Answers:

\(\displaystyle x=\frac{1}{3}\)

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

\(\displaystyle x=\frac{1}{3},2\)

\(\displaystyle x=-2,-9\)

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

Correct answer:

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

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 \(\displaystyle -\frac{1}{3}\) and \(\displaystyle 2\)

Example Question #5 : Solving Quadratic Equations

Find the zeros.

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

Possible Answers:

\(\displaystyle x=9,-9\)

\(\displaystyle x=\frac{9}{5},-\frac{9}{5}\)

\(\displaystyle x=-\frac{9}{5}\)

\(\displaystyle x=\frac{9}{5}\)

\(\displaystyle x=5\)

Correct answer:

\(\displaystyle x=\frac{9}{5},-\frac{9}{5}\)

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 \(\displaystyle \frac{9}{5}\) and \(\displaystyle -\frac{9}{5}\).

Example Question #311 : Algebra Ii

Find the zeros. 

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

Possible Answers:

\(\displaystyle x=8\)

\(\displaystyle x=\frac{1}{4}\)

\(\displaystyle x=0,\frac{1}{4}\)

\(\displaystyle x=0\)

\(\displaystyle x=0,4\)

Correct answer:

\(\displaystyle x=0,\frac{1}{4}\)

Explanation:

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

Example Question #6 : Solving Quadratic Equations

Find the zeros. 

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

Possible Answers:

\(\displaystyle x=3,2\)

\(\displaystyle x=-3,3\)

\(\displaystyle x=-3,-2\)

\(\displaystyle x=2\)

\(\displaystyle x=4,2\)

Correct answer:

\(\displaystyle x=-3,-2\)

Explanation:

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

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

which can be further factored to

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

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

Example Question #1 : Solving Quadratic Equations

Find the zeros. 

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

Possible Answers:

\(\displaystyle x=-6,6\)

\(\displaystyle x=1,-1\)

\(\displaystyle x=0\)

\(\displaystyle x=0,6\)

\(\displaystyle x=6\)

Correct answer:

\(\displaystyle x=0,6\)

Explanation:

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

Example Question #1 : Solving Quadratic Equations

Find the zeros.

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

Possible Answers:

\(\displaystyle x=4\)

\(\displaystyle x=0,2,-2\)

\(\displaystyle x=2,-4\)

\(\displaystyle x=-2,4\)

\(\displaystyle x=2\)

Correct answer:

\(\displaystyle x=2,-4\)

Explanation:

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

Example Question #7 : Solving Quadratic Equations

Find the zeros. 

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

Possible Answers:

\(\displaystyle x=-5,8\)

\(\displaystyle x=0\)

\(\displaystyle x=5,8\)

\(\displaystyle x=-5\)

\(\displaystyle x=-8\)

Correct answer:

\(\displaystyle x=-5,8\)

Explanation:

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

Learning Tools by Varsity Tutors