DERIVING THE QUADRATIC FORMULA
ax2 + bx + c = 0
Subtract c from both sides of the equation.
ax2 + bx = -c
Divide both sides by the coefficient of x2.
x2 + (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.
x2 + (b/a)x + b2/4a2 = -c/a + b2/4a2
Simplify the left side into a square.
(x + b/2a)2 = -c/a + b2/4a2
Simplify the right side by finding the lowest common denominator.
(x + b/2a)2 = -4ac/4a2 + b2/4a2
Add the fractions on the right.
(x + b/2a)2 = (b2 - 4ac)/4a2
Take the square root of both sides.
Note: The term +/-sqrt = plus or minus the square root of.
x + b/2a = +/-sqrt(b2 - 4ac)/2a
Subtract b/2a from both sides and simplify.
x = [-b +/-sqrt(b2 - 4ac)]/2a
AN EXAMPLE USING THE QUADRATIC FORMULA
2x2 + 3x + 22 = 49
2x2 + 3x + (-27) = 0
Using the quadratic formula...
x = [-3 +/-sqrt(32 - 4(2)(-27))]/[2(2)]
x = [-3 +/-sqrt(225)]/4
x = -3/4 +/-(15/4)
x = 3 or x = -4.5
This tutorial created and designed by
Curtis Lee Hall