Solving Quadratic Equations Using the Quadratic Formula Summary

Key Concepts

• Quadratic Formula The solutions to a quadratic equation of the form \(a{x}^{2}+bx+c=0,\)\(a\ne 0\) are given by the formula:
  \(x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}\)
• Solve a Quadratic Equation Using the Quadratic Formula
  To solve a quadratic equation using the Quadratic Formula.
  1. Write the quadratic formula in standard form. Identify the \(a,b,c\) values.
  2. Write the quadratic formula. Then substitute in the values of \(a,b,c.\)
  3. Simplify.
  4. Check the solutions.
• Using the Discriminant, \({b}^{2}-4ac\), to Determine the Number of Solutions of a Quadratic Equation
  For a quadratic equation of the form \(a{x}^{2}+bx+c=0,\)\(a\ne 0,\)
  • if \({b}^{2}-4ac>0\), the equation has 2 solutions.
  • if \({b}^{2}-4ac=0\), the equation has 1 solution.
  • if \({b}^{2}-4ac<0\), the equation has no real solutions.
• To identify the most appropriate method to solve a quadratic equation:
  1. Try Factoring first. If the quadratic factors easily this method is very quick.
  2. Try the Square Root Property next. If the equation fits the form \(a{x}^{2}=k\) or \(a{\left(x-h\right)}^{2}=k\), it can easily be solved by using the Square Root Property.
  3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Glossary

discriminant

In the Quadratic Formula, \(x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}\) the quantity \({b}^{2}-4ac\) is called the discriminant.