Quadratics

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

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


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

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

OR

\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

Calculator Click What You See Comment
clear blinker clears screen
4 × 3 square – 18 × 3 + 20 4*3²-18*3+20 y = 4x² – 18x + 20
Plug in 3 for x.
enter 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

Calculator Click What You See Comment
clear blinker clears screen
4 × ( -3 ) square
– 18 × -3 + 20
4*(-3)²-18*-3+20 y = 4x² – 18x + 20
Plug in -3 for x.

Parentheses protect order of operations.
enter 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
\bf\displaystyle x =\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

OR

\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}

Calculator Click What You See Comment
clear blinker clears screen
fraction – -9 +
2nd square root ( -9 ) square
– 4 × 2 × 4 down 2 × 2 right
\frac{--9+\sqrt{{{(-9)}^{2}}-4\times 2\times 4}}{2\times 2} x=\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}
enter 4 Answer
fraction – -9 –
2nd square root ( -9 ) square
– 4 × 2 × 4 down 2 × 2 right
\frac{--9-\sqrt{{{(-9)}^{2}}-4\times 2\times 4}}{2\times 2} x=\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}
enter \bf\displaystyle\frac{1}{2} Answer
Quadratics


Practice – Questions
1.  For the quadratic equation y = x² + 6x + 8, solve for y when x = 2.

2.  For the quadratic equation y = x² + 6x + 8, solve for y when x = -2.

3.  For the quadratic equation y = 6x² – 18x + 12, solve for y when x = 4.

4.  For the quadratic equation y = 6x² – 18x + 12, solve for y when x = -4.

5.  For the quadratic equation 3x² – 7x + 4 = 0, solve for x.


Practice – Answers
1.  y = 24

2.  y = 0

3.  y = 36

4.  y = 180

5.  x = 1  or  x = \bf\displaystyle\frac{4}{3}

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