Solving Quadratic Equations by Square Roots
If the quadratic equation only involves a variable expression that is squared, then we can solve it using square roots. This would be an equation in the form \(ax^2+c=0\) or \(a\left(x-b\right)^2+c=0\).
Steps to Solving Quadratic Equations by Square Roots:
When solving an equation such as \(x^2=9\), we need to remember that both \(x=3\) and \(x=-3\) will be solutions. In general, the solutions to the equation \(x^2=a\) is \(x=\pm\sqrt{a}\).
Solve the following equations.
Example 1: Solve \(-3x^2+7=-47\).
If the quadratic equation only involves a variable expression that is squared, then we can solve it using square roots. This would be an equation in the form \(ax^2+c=0\) or \(a\left(x-b\right)^2+c=0\).
Steps to Solving Quadratic Equations by Square Roots:
- Isolate the squared term/expression.
- Take the square root of both sides of the equation.
- Simplify and continue solving if necessary.
When solving an equation such as \(x^2=9\), we need to remember that both \(x=3\) and \(x=-3\) will be solutions. In general, the solutions to the equation \(x^2=a\) is \(x=\pm\sqrt{a}\).
Solve the following equations.
Example 1: Solve \(-3x^2+7=-47\).
\(\begin{align}-3x^2&=-54\ & \ \ &\text{1) Isolate the squared term.}\\x^2&=18\ & \ \ &\text{2) Take the square root of both sides.}\\x&=\pm\sqrt{18}\ & \ \ &\text{3) Simplify.}\\x&=\pm3\sqrt{2}\end{align}\)
Example 2: Solve \(5x^2+30=0\).
\(\begin{align}5x^2&=-30\ & \ \ &\text{1) Isolate the squared term.}\\x^2&=-6\ & \ \ &\text{2) Take the square root of both sides.}\\x&=\pm\sqrt{-6}\ & \ \ &\text{3) Simplify.}\\x&=\pm{i}\sqrt6\end{align}\)
Example 3: Solve \(2\left(x+8\right)^2-43=57\).
\(\begin{align}2\left(x+8\right)^2&=100\ & \ \ &\text{1) Isolate the squared term.}\\
\left(x+8\right)^2&=50\ & \ \ &\text{2) Take the square root of both sides.}\\
x+8&=\pm\sqrt{50}\ & \ \ &\text{3) Simplify.}\\
x&=-8\pm5\sqrt{2}\ & \ \ &\text{4) Finish solving...note that the -8 and}\ 5\sqrt{2}\ \text{are not like terms and cannot be combined.}\end{align}\)
\left(x+8\right)^2&=50\ & \ \ &\text{2) Take the square root of both sides.}\\
x+8&=\pm\sqrt{50}\ & \ \ &\text{3) Simplify.}\\
x&=-8\pm5\sqrt{2}\ & \ \ &\text{4) Finish solving...note that the -8 and}\ 5\sqrt{2}\ \text{are not like terms and cannot be combined.}\end{align}\)
Example 4: Solve \(3\left(x-5\right)^2-15=17\).
Quick Check
Solve the following equations.
1) \(-\frac{2}{3}x^2+1=-29\)
2) \(\left(x+7\right)^2-5=-25\)
Quick Check Solutions
Solve the following equations.
1) \(-\frac{2}{3}x^2+1=-29\)
2) \(\left(x+7\right)^2-5=-25\)
Quick Check Solutions