Quadratic Formula Practice Worksheet

Practice solving quadratic equations using the quadratic formula with step-by-step solutions.

Use the quadratic formula to solve each equation. Simplify your answers whenever possible.

Quadratic Formula

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

Level 1: Basic Quadratics

Problem 1
\[ x^2 + 5x + 6 = 0 \]
Show solution

Here: \(a=1\), \(b=5\), \(c=6\)

\[ x= \frac{ -5 \pm \sqrt{5^2-4(1)(6)} }{ 2(1) } \]
\[ x= \frac{ -5 \pm \sqrt{25-24} }{ 2 } \]
\[ x= \frac{ -5 \pm 1 }{ 2 } \]
\[ x=-2,\ -3 \]
Problem 2
\[ x^2 - 7x + 10 = 0 \]
Show solution

Here: \(a=1\), \(b=-7\), \(c=10\)

\[ x= \frac{ 7 \pm \sqrt{(-7)^2-4(1)(10)} }{ 2 } \]
\[ x= \frac{ 7 \pm \sqrt{49-40} }{ 2 } \]
\[ x= \frac{ 7 \pm 3 }{ 2 } \]
\[ x=5,\ 2 \]
Problem 3
\[ x^2 + 4x - 12 = 0 \]
Show solution

Here: \(a=1\), \(b=4\), \(c=-12\)

\[ x= \frac{ -4 \pm \sqrt{4^2-4(1)(-12)} }{ 2 } \]
\[ x= \frac{ -4 \pm \sqrt{16+48} }{ 2 } \]
\[ x= \frac{ -4 \pm 8 }{ 2 } \]
\[ x=2,\ -6 \]

Level 2: Harder Quadratics

Problem 4
\[ 2x^2 + 3x - 2 = 0 \]
Show solution

Here: \(a=2\), \(b=3\), \(c=-2\)

\[ x= \frac{ -3 \pm \sqrt{3^2-4(2)(-2)} }{ 2(2) } \]
\[ x= \frac{ -3 \pm \sqrt{9+16} }{ 4 } \]
\[ x= \frac{ -3 \pm 5 }{ 4 } \]
\[ x=\frac{1}{2},\ -2 \]
Problem 5
\[ 3x^2 - 5x - 2 = 0 \]
Show solution

Here: \(a=3\), \(b=-5\), \(c=-2\)

\[ x= \frac{ 5 \pm \sqrt{(-5)^2-4(3)(-2)} }{ 6 } \]
\[ x= \frac{ 5 \pm \sqrt{25+24} }{ 6 } \]
\[ x= \frac{ 5 \pm 7 }{ 6 } \]
\[ x=2,\ -\frac{1}{3} \]
Problem 6
\[ x^2 - 2x - 15 = 0 \]
Show solution
\[ x= \frac{ 2 \pm \sqrt{(-2)^2-4(1)(-15)} }{ 2 } \]
\[ x= \frac{ 2 \pm \sqrt{4+60} }{ 2 } \]
\[ x= \frac{ 2 \pm 8 }{ 2 } \]
\[ x=5,\ -3 \]

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