Quadratics

Basics
A quadratic is an equation or formula in which a variable is squared.
(In the equation above, the variable x is squared.)

A question involving quadratics could involve a relatively simple plug-in, where you plug in a number for x to solve for y using the quadratic equation.

A question involving quadratics could involve more complicated calculations, where you must solve for x using the quadratic formula.

Because the quadratic formula features ± calculations, in most cases you will produce two answers for x.

When squaring a negative number, remember to use parentheses to protect the order of operations.

Formulas (Mathematics Formula Sheet)
standard form of a quadratic equation
y = $\bf\displaystyle\ a{{x}^{2}}+bx+c$

quadratic formula
x = $\bf\displaystyle\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

Formulas (NOT on Mathematics Formula Sheet)
quadratic formula
x = $\bf\displaystyle\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ means —>
“

$\bf\displaystyle x=\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$

$\bf\displaystyle x=\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$

Question
For the quadratic equation y = 4x² – 18x + 20, solve for y when x = 3.

Answer
y = 2

Answer Process
See Plug-In.
y = 4x² – 18x + 20
y = 4 × 3² – 18 × 3 + 20 = 2

Input	Display	Comment
	blinker	clears screen

4 × 3 – 18 × 3 + 20	43²-183+20	y = 4x² – 18x + 20 Plug in 3 for x.

	2	Answer

		Quadratics

Question
For the quadratic equation y = 4x² – 18x + 20, solve for y when x = -3.

Answer
y = 110

Answer Process
See Plug-In.
y = 4x² – 18x + 20
y = 4 × (-3)² – 18 × -3 + 20 = 110

Input	Display	Comment
	blinker	clears screen

4 × -3 – 18 × -3 + 20	4(-3)²-18-3+20	y = 4x² – 18x + 20 Plug in -3 for x. Parentheses protect order of operations.

	110	Answer

		Quadratics

Question
For the quadratic equation 2x² – 9x + 4 = 0, solve for x.

Answer
x = $\bf\displaystyle\frac{1}{2}$ and x = 4

Answer Process
quadratic formula
x = $\bf\displaystyle\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
ax² + bx + c = 0
2x² – 9x + 4 = 0
a = 2
b = -9
c = 4

$\bf\displaystyle x=\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$

$\bf\displaystyle x=\frac{--9+\sqrt{{{(-9)}^{2}}-4\times 2\times 4}}{2\times 2}$ = 4

$\bf\displaystyle x=\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$

$\bf\displaystyle x=\frac{--9-\sqrt{{{(-9)}^{2}}-4\times 2\times 4}}{2\times 2}$ = $\bf\displaystyle\frac{1}{2}$

Input	Display	Comment
	blinker	clears screen

$fraction$ – -9 + -9 – 4 × 2 × 4 2 × 2	$\bf\displaystyle x=\frac{--9+\sqrt{{{(-9)}^{2}}-4\times 2\times 4}}{2\times 2}$	$\bf\displaystyle x=\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$

	4	Answer

$fraction$ – -9 – -9 – 4 × 2 × 4 2 × 2	$\bf\displaystyle x=\frac{--9-\sqrt{{{(-9)}^{2}}-4\times 2\times 4}}{2\times 2}$	$\bf\displaystyle x=\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$

	$\bf\displaystyle\frac{1}{2}$	Answer

		Quadratics