What is the quadratic formula used for?

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1 Answer
seph
Oct 19, 2014

The quadratic formula is used to get the roots of a quadratic equation, if the roots exists at all.

We usually just perform factorization to get the roots of a quadratic equation. However, this is not always possible (especially when the roots are irrational)

The quadratic formula is

#x = (-b +- root 2 (b^2 - 4ac))/(2a)#

Example 1:

#y = x^2 -3x - 4#
#0 = x^2 -3x - 4#

#=> 0 = (x - 4)(x + 1)#
#=> x = 4, x = -1#

Using the quadratic formula, let's try to solve the same equation

#x = (-(-3) +- root 2 ((-3)^2 - 4*1*(-4)))/(2 * 1)#
#=> x = (3 +- root 2 (9 + 16))/2#
#=> x = (3 +- root 2 (25))/2#
#=> x = (3 + 5)/2, x = (3 - 5)/2#
#=> x = 4, x = -1#

Example 2:

#y = 2x^2 -3x - 5#
#0 = 2x^2 - 3x - 5#

Performing factorization is a little hard for this equation, so let's jump straight to using the quadratic formula

#x = (-(-3) +- root 2 ((-3)^2 - 4 * 2 * (-5)))/(2 * 2)#

#x = (3 +- root 2 (9 + 40))/4#

#x = (3 +- root 2 49)/4#

#x = (3 + 7)/4, x = (3 - 7)/4#

#x = 5/2, x = -1#

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