About Quadratic in Form:
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.
$$a)\hspace{.2em}5x^4 + 25x^2=-30$$
$$b)\hspace{.2em}3x^4=24x^2 - 21$$
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#2:
Instructions: solve each equation.
$$a)\hspace{.2em}2x^4 + 20x^2 + 10=-2x^2 - 10$$
$$b)\hspace{.2em}x - 3 - 7\sqrt{x - 3}=-10$$
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#3:
Instructions: solve each equation.
$$a)\hspace{.2em}x + 48=14 \sqrt{x}$$
$$b)\hspace{.2em}x + 16=10 \sqrt{x}$$
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#4:
Instructions: solve each equation.
$$a)\hspace{.2em}19x^{\frac{2}{3}}+ 34x^{\frac{1}{3}}+ 8=-2x^{\frac{2}{3}}$$
$$b)\hspace{.2em}7x^{\frac{2}{3}}- 38x^{\frac{1}{3}}- 17=-5 + 2x^{\frac{1}{3}}$$
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#5:
Instructions: solve each equation.
$$a)\hspace{.2em}x^4 - 2x^2 + 1 - 12=-(x^2 - 1)$$
$$b)\hspace{.2em}x^2 + 4x + 4=-11(x + 2) + 12$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}x=\pm i\sqrt{2}, \pm i \sqrt{3}$$
$$b)\hspace{.2em}x=\pm \sqrt{7}, \pm 1$$
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#2:
Solutions:
$$a)\hspace{.2em}x=\pm i, \pm i \sqrt{10}$$
$$b)\hspace{.2em}x=7, 28$$
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#3:
Solutions:
$$a)\hspace{.2em}x=36, 64$$
$$b)\hspace{.2em}x=4, 64$$
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#4:
Solutions:
$$a)\hspace{.2em}x=-\frac{64}{27}, -\frac{8}{343}$$
$$b)\hspace{.2em}x=-\frac{8}{343}, 216$$
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#5:
Solutions:
$$a)\hspace{.2em}x=\pm 2, \pm i\sqrt{3}$$
$$b)\hspace{.2em}x=-14, -1$$
