Solve Quadratic Equations with the Quadratic Formula
The quadratic formula can be used to solve any quadratic equation in the form \(ax^2+bx+c=0\). The quadratic formula can be derived by completing the square on \(ax^2+bx+c\). Watch the video below to see the derivation.
The quadratic formula can be used to solve any quadratic equation in the form \(ax^2+bx+c=0\). The quadratic formula can be derived by completing the square on \(ax^2+bx+c\). Watch the video below to see the derivation.
The quadratic formula is \(x=\large\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
While this will work with any quadratic equation, it is often not the most efficient because of the simplification that is necessary. If possible, try to use one of the other methods first.
Steps to Solving Quadratic Equations using the Quadratic Formula:
- Write the equation in the form \(ax^2+bx+c=0\).
- Identify the values for \(a\), \(b\), and \(c\) and substitute into the quadratic formula.
- Simplify as needed.
Solve the following equations.
Example 1: Solve \(4x^2-8x=5\).
\(\begin{align}&4x^2-8x-5=0\ & \ &\text{1) Write in standard form.}\\&a=4, b=-8, \text{and}\ c=-5\ & \ &\text{2) Identify the values for a, b, and c.}\\&x=\frac{-\left(-8\right)\pm\sqrt{\left(-8\right)^2-4\left(4\right)\left(-5\right)}}{2\left(4\right)}\ & \ &\text{3) Substitute values of a, b and c into the quadratic formula.}\\&x=\frac{8\pm\sqrt{144}}{8}\ & \ &\text{4) Simplify each term in the expression.}\\&x=\frac{8\pm12}{8}\ & \ &\text{5) Simplify the square root.}\\&x=\frac{8+12}{8}\ \text{or}\ x=\frac{8-12}{8}\ & \ &\text{6) Rewrite into two expressions (one with addition in the numerator}.\\
&\ & \ &\text{and one with subtraction).}\\&x=\frac{5}{2},-\frac{1}{2}\ & \ &\text{7)
Simplify.}\end{align}\)
&\ & \ &\text{and one with subtraction).}\\&x=\frac{5}{2},-\frac{1}{2}\ & \ &\text{7)
Simplify.}\end{align}\)
Example 2: Solve \(3x^2+7=-6x\).
\(\begin{align}&3x^2+6x+7=0\ & \ &\text{1) Write in standard form.}\\&a=3, b=6, \text{and}\ c=7\ & \ &\text{2) Identify the values for a, b, and c.}\\&x=\frac{-6\pm\sqrt{6^2-4\left(3\right)\left(7\right)}}{2\left(3\right)}\ & \ &\text{3) Substitute values of a, b and c into the quadratic formula.}\\&x=\frac{-6\pm\sqrt{-48}}{6}\ & \ &\text{4) Simplify each term in the expression.}\\&x=\frac{-6\pm4i\sqrt{3}}{6}\ & \ &\text{5) Simplify the sqare root.}\\&x=\frac{-3\pm2i\sqrt{3}}{3}\ & \ &\text{6) Divide each term by 2. Since the numerator involves both a rational and irrational term,}\\
&\ & \ &\text{there is no further simplification that needs to be done.}\end{align}\)
&\ & \ &\text{there is no further simplification that needs to be done.}\end{align}\)
Example 3: Solve \(-2x^2+7x+6=0\).