We will encounter non-quadratic equations that are quadratic in form. We can create a quadratic equation by making a simple substitution. We can then solve the quadratic equation using the quadratic formula. When done, we substitute once more to obtain a solution in terms of the original variable involved.

Test Objectives
• Demonstrate the ability to identify a non-quadratic equation which is quadratic in form
• Demonstrate the ability to use substitution to create a quadratic equation
• Demonstrate the ability to solve an equation which is quadratic in form

#1:

Instructions: solve each equation.

$$a)\hspace{.2em}5x^4 + 25x^2=-30$$

$$b)\hspace{.2em}3x^4=24x^2 - 21$$

#2:

Instructions: solve each equation.

$$a)\hspace{.2em}2x^4 + 20x^2 + 10=-2x^2 - 10$$

$$b)\hspace{.2em}4 + 3 \sqrt{x}=7$$

#3:

Instructions: solve each equation.

$$a)\hspace{.2em}x + 48=14 \sqrt{x}$$

$$b)\hspace{.2em}x + 16=10 \sqrt{x}$$

#4:

Instructions: solve each equation.

$$a)\hspace{.2em}19x^{\frac{2}{3}}+ 34x^{\frac{1}{3}}+ 8=-2x^{\frac{2}{3}}$$

$$b)\hspace{.2em}7x^{\frac{2}{3}}- 38x^{\frac{1}{3}}- 17=-5 + 2x^{\frac{1}{3}}$$

#5:

Instructions: solve each equation.

$$a)\hspace{.2em}x^4 - 2x^2 + 1 - 12=-(x^2 - 1)$$

$$b)\hspace{.2em}x^2 + 4x + 4=-11(x + 2) + 12$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x=\pm i\sqrt{2}, \pm i \sqrt{3}$$

$$b)\hspace{.2em}x=\pm \sqrt{7}, \pm 1$$

#2:

Solutions:

$$a)\hspace{.2em}x=\pm i, \pm i \sqrt{10}$$

$$b)\hspace{.2em}x=1$$

#3:

Solutions:

$$a)\hspace{.2em}x=36, 64$$

$$b)\hspace{.2em}x=4, 64$$

#4:

Solutions:

$$a)\hspace{.2em}x=-\frac{64}{27}, -\frac{8}{343}$$

$$b)\hspace{.2em}x=-\frac{8}{343}, 216$$

#5:

Solutions:

$$a)\hspace{.2em}x=\pm 2, \pm i\sqrt{3}$$

$$b)\hspace{.2em}x=-14, -1$$