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Line 9: − You can prove the formula the following way:+ + + + + + + Line 25:
Line 31: − This method of deriving the formula is done via [[completing the square]].+ − −
Simplicity isn't what's important. What's important is that one way is guessing and the other is solving.
The '''quadratic formula''' can be used when simpler methods of solving a [[quadratic equation]] do not work.
The '''quadratic formula''' is the formula for finding the solutions of [[quadratic equation]]s. Students are sometimes taught a method known as ''factoring'', but that's really just looking for a lucky guess.
First, the quadratic equation must be reduced to this format:
First, the quadratic equation must be reduced to this format:
:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>
:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>
The formula is derived in the following way, which is known as [[completing the square]]:
:<math>ax^2+bx+c=0\!</math>
:<math>ax^2+bx+c=0\!</math>
:<math>x^2+\frac{b}{a}x+\frac{c}{a}=0\!</math>
:<math>x^2+\frac{b}{a}x+\frac{c}{a}=0\!</math>
::Now we need to get something of the form <math>(x+Q)^2\!</math> that matches the first two terms. We have
:::<math>(x+Q)^2 = x^2 + 2Qx + Q^2\!</math>
::So we need <math>Q = \frac{b}{2a}\!</math> to get a match.
:::<math>(x+\frac{b}{2a})^2 = x^2 + \frac{b}{a}x + (\frac{b}{2a})^2\!</math>
::Plugging that in, we get
:<math>(x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\!</math>
:<math>(x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\!</math>
:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>
:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>
You can check that the formula is correct by substituting the formula (with either sign for the square root) in place of '''x''' in <math>ax^2+bx+c=0\!</math> and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.
You can assert that the formula is correct by substituting the formula in place of '''x''' in <math>ax^2+bx+c=0\!</math> and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.
[[Category:Mathematics]]
[[Category:Mathematics]]