Quadratic equations



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Quadratic Equations

- Solve a quadratic equation using square roots. (Worksheet)
- Solve a quadratic equation by factoring (Worksheet)
- Solve a quadratic equation using the quadratic formula. (Worksheet)
- Using the discriminant
- Equations in Quadratic Form
- Problems
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Equations in Quadratic Form

Some equations are not quadratic equations, but are in the same form.
To solve equations such as that, make a substitution, solve for the new variable, and then solve for the original variable

$\fs2x^4-5x^2+4\;=\;0\;\Rightarrow\;\;y=x^2$
$\fs2y^2-5y+4\;=\;0\;\Rightarrow\;y=\frac{5\pm\sqrt{(-5)^2-4\;1\;4}}{2\;1}\;\Rightarrow\;y=\frac{5\pm\sqrt{25-16}}{2}\;\Rightarrow\;y=\frac{5\pm3}{2}$

$\fs2y_1=\frac{5+3}{2}=\frac{8}{2}=4\;\;\;\;y_2=\frac{5-3}{2}=\frac{1}{2}=1$

These give

$\fs2x^2\;=\;y_1\;\Rightarrow\;\;x^2\;=\;4\;\Rightarrow\;\;x\;=\;\pm2$

$x^2\;=\;y_2\;\Rightarrow\;x^2\;=\;1\;\Rightarrow\;\;x\;=\;\pm1\;\;\;$

Solve $\fs1x^4-5x^2+4\;=\;0$

No solution
One solution x=
Two solutions x₁= x₂=
Three solutions x₁= x₂= x₃=
Four solutions x₁= x₂= x₃= x₄=