How to Solve a Quadratic Equation? (+FREE Worksheet!)

Related Topics

• How to Graph Quadratic Functions
• How to Solve Quadratic Inequalities
• How to Graph Quadratic Inequalities

Step by step guide to Solving a Quadratic Equation

1. Write the equation in the form of: \(ax^2+bx+c=0\)
2. Factorize the quadratic and solve for the variable.
3. Use quadratic formula if you couldn’t factorize the quadratic.
4. Quadratic formula: \( \color{blue}{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}\)

Solving a Quadratic Equation – Example 1:

Find the solutions of each quadratic. \(x^2+7x+10=0\)

Answer:

\(x^2+7x+10=0\)

You can use factorization method. \(x^2+7x+10=0\)

\((x+5)(x+2)=0\)
Then: \((x=-5)\) and \((x=-2)\)
You can also use quadratic formula: \(=\frac{-b±\sqrt{b^2-4ac}}{2a} , a=1,b=7\) and \(c=10\), then: \(x=\frac{-7±\sqrt{7^2-(4)×(1)×(10)}}{2×1}\)
\(x_{1}=\frac{-7+\sqrt{7^2-(4) × (1) × (10)}}{2 × 1}= \frac{-7+\sqrt{49-40}}{2}= \frac{-7+\sqrt{9}}{2} = \frac{-7+3}{2} =\frac {-4}{2}= -2\) ,

\(x_{2}=\frac{-7-\sqrt{7^2-(4) × (1) × (10)}}{2 × 1}= \frac{-7-\sqrt{49-40}}{2}= \frac{-7-\sqrt{9}}{2} = \frac{-7-3}{2} =\frac {-10}{2}= -5\)

Solving a Quadratic Equation – Example 2:

Find the solutions of each quadratic. \(x^2+4x+3=0\)

Answer:

Use quadratic formula: \(=\frac{-b±\sqrt{b^2-4ac}}{2a} , a=1,b=4\) and \(c=3 \), then: \(x=\frac{-4±\sqrt{4^2-(4) × (1) × (3)}}{2 × 1}\)

\(x_{1}=\frac{-4+\sqrt{4^2-(4) × (1) × (3)}}{2 × 1}= \frac{-4+\sqrt{16-12}}{2}= \frac{-4+\sqrt{4}}{2} = \frac{-4+2}{2} =\frac {-2}{2}= -1\),

\(x_{2}=\frac{-4-\sqrt{4^2-(4) × (1) × (3)}}{2 × 1}= \frac{-4-\sqrt{16-12}}{2}= \frac{-4-\sqrt{4}}{2} = \frac{-4-2}{2} =\frac {-6}{2}= -3\)

Solving a Quadratic Equation – Example 3:

Find the solutions of each quadratic. \(x^2+5x-6\)

Answer:

Use quadratic formula: \(=\frac{-b±\sqrt{b^2-4ac}}{2a} , a=1,b=5\) and \(c=-6\) , then: \(x=\frac{-5±\sqrt{5^2-(4) × (1) × (-6)}}{2 × 1}\) ,

\(x_{1}=\frac{-5+\sqrt{5^2-(4) × (1) × (-6)}}{2 × 1}= \frac{-5+\sqrt{25+24}}{2}= \frac{-5+\sqrt{49}}{2} = \frac{-5+7}{2} =\frac {2}{2}= 1\),

\(x_{2}=\frac{-5-\sqrt{5^2-(4) × (1) × (-6)}}{2 × 1}= \frac{-5-\sqrt{25+24}}{2}= \frac{-5-\sqrt{49}}{2} = \frac{-5-7}{2} =\frac {-12}{2}= -6\)

Solving a Quadratic Equation – Example 4:

Find the solutions of each quadratic. \(x^2+6x+8\)

Answer:

Use quadratic formula: \(=\frac{-b±\sqrt{b^2-4ac}}{2a} , a=1,b=6\) and \(c=8\), then: \(x=\frac{-6±\sqrt{6^2-(4) × (1) × (8)}}{2 × 1}\) ,

\(x_{1}=\frac{-6+\sqrt{6^2-(4) × (1) × (8)}}{2 × 1}= \frac{-6+\sqrt{36-32}}{2}= \frac{-6+\sqrt{4}}{2} = \frac{-6+2}{2} =\frac {-4}{2}= -2\),

\(x_{2}=\frac{-6-\sqrt{6^2-(4) × (1) × (8)}}{2 × 1}= \frac{-6-\sqrt{36-32}}{2}= \frac{-6-\sqrt{4}}{2} = \frac{-6-2}{2} =\frac {-8}{2}= -4\)

Exercises for Solving a Quadratic Equation

Solve each equation.

• \(\color{blue}{x^2-5x-14=0}\)
• \(\color{blue}{x^2+8x+15=0}\)
• \(\color{blue}{x^2-5x-36=0}\)
• \(\color{blue}{x^2-12x+35=0}\)
• \(\color{blue}{x^2+12x+32=0}\)
• \(\color{blue}{5x^2+27x+28=0}\)

Download Solving a Quadratic Equation Worksheet

• \(\color{blue}{x=-2,x=7}\)
• \(\color{blue}{x=-3,x=-5}\)
• \(\color{blue}{x=9,x=-4}\)
• \(\color{blue}{x=7,x=5}\)
• \(\color{blue}{x=-4,x=-8}\)
• \(\color{blue}{x=-\frac{7}{5},x=-4}\)