By Alwin Peng
The quadratic formula is used in order to find the value of x in an equation such as ax^2 + bx + c = 0.
It is x = (-b + (b^2 – 4ac)^(1/2))/2a.
It simplifies a more complicated way to solve the expression. Below, I will show you why the formula works by explaining how it was derived.Deriving the quadratic formula goes as follows:
ax^2 + bx + c = 0
ax^2 + bx = -c .We subtract both sides by C
x^2 + bx/a = -c/a .We divide both sides by A
x^2 + bx/a + (b^2)/(4a^2) = -c/a + (b^2)/(4a^2) .We add (b^2)/(4a^2) to both sides so that the left side of the equation is factorable
(x + b/2a)^2 = -c/a + (b^2)/(4a^2) We factor the left side
(x + b/2a)^2 = -4ac/(4a^2) + (b^2)/(4a^2) We multiply the top and bottom of the fraction -c/a by 4a so that it can be added with (b^2)/(4a^2), achieving the common denominator of 4a^2 in both fractions.
(x + b/2a)^2 = (-4ac + (b^2))/(4a^2) We now add the like terms
x + b/2a = (((b^2) – 4ac)^(1/2))/2a We square root both sides
At last, we subtract both sides by b/2a, isolating x and getting the quadratic formula:
x = (-b + (b^2 – 4ac)^(1/2))/2a
The method that we just used to obtain the quadratic formula is known as “completing the square”, but just plugging values into the quadratic formula is a faster and simplified way to complete the square.