How to Solve a Quadratic Equation? (+FREE Worksheet!)
Learn how to simplify and solve a Quadratic Equation in a few simple and easy steps.
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
- Write the equation in the form of: \(ax^2+bx+c=0\)
- Factorize the quadratic and solve for the variable.
- Use quadratic formula if you couldn’t factorize the quadratic.
- Quadratic formula: \( \color{blue}{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}\)
Best Algebra Prep Resource
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 the 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: \(x=\frac{-5±\sqrt{5^2-(4) × (1) × (-6)}}{2 × 1}\) ,
\(x_{1}=\frac{-5+\sqrt{5^2-(4) × (1) × (-6)}}{2 × 1}= \frac{-5+\sqrt{25+24}}{2}= \frac{-5+\sqrt{49}}{2} = \frac{-5+7}{2} =\frac {2}{2}= 1\),
\(x_{2}=\frac{-5-\sqrt{5^2-(4) × (1) × (-6)}}{2 × 1}= \frac{-5-\sqrt{25+24}}{2}= \frac{-5-\sqrt{49}}{2} = \frac{-5-7}{2} =\frac {-12}{2}= -6\)
A Perfect Practice Workbook for Algebra I
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
- \(\color{blue}{x=-2,x=7}\)
- \(\color{blue}{x=-3,x=-5}\)
- \(\color{blue}{x=9,x=-4}\)
- \(\color{blue}{x=7,x=5}\)
- \(\color{blue}{x=-4,x=-8}\)
- \(\color{blue}{x=-\frac{7}{5},x=-4}\)
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