**DERIVING THE QUADRATIC FORMULA**

ax^{2} + bx + c = 0

Subtract c from both sides of the equation.

ax^{2} + bx = -c

Divide both sides by the coefficient of x^{2}.

x^{2} + (b/a)x = -c/a

Divide the coefficient of x by 2, square it, then add to both sides.

Note: This makes a perfect square on the left side.

x^{2} + (b/a)x + b^{2}/4a^{2} = -c/a + b^{2}/4a^{2}

Simplify the left side into a square.

(x + b/2a)^{2} = -c/a + b^{2}/4a^{2}

Simplify the right side by finding the lowest common denominator.

(x + b/2a)^{2} = -4ac/4a^{2} + b^{2}/4a^{2}

Add the fractions on the right.

(x + b/2a)^{2} = (b^{2} - 4ac)/4a^{2}

Take the square root of both sides.

Note: The term +/-sqrt = plus or minus the square root of.

x + b/2a = +/-sqrt(b^{2} - 4ac)/2a

Subtract b/2a from both sides and simplify.

x = [-b +/-sqrt(b^{2} - 4ac)]/2a
**AN EXAMPLE USING THE QUADRATIC FORMULA**

If...

2x^{2} + 3x + 22 = 49

Then...

2x^{2} + 3x + (-27) = 0

Using the quadratic formula...

x = [-3 +/-sqrt(3^{2} - 4(2)(-27))]/[2(2)]

x = [-3 +/-sqrt(225)]/4

x = -3/4 +/-(15/4)

Therefore...

x = 3 or x = -4.5

This tutorial created and designed by

Curtis Lee Hall

www.InTheBeginning.com