Raymond B. answered • 12/16/20
Math, microeconomics or criminal justice
ax^2 + bx + c =0 is the standard form equation for a quadratic
divide by a to get
x^2 + (b/a)x + c/a =0
move c/a to the right side
x^2 + (b/a) x = -c/a
complete the square by taking half the x coefficient and squaring it, then add it to both sides
x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a
change c/a to 4ac/4a^2
x^2 + (b/a)x + (b/2a)^2 = b^2/4a^2 - 4ac/4a^2 combine the last two terms to get (b^2- - 4ac)/4a^2
x^2 + (b/a)x + (b/2a)^2 = (b^2 -4ac)/4a^2
(x+ b/2a)^2 = (b^2-4ac)/4a^2
take square roots of both sides
x+b/2a = +(1/2a)sqr(b^2-4ac)
x = -b/2a + (1/2a)sqr(b^2-4ac)
combine the right side under one denominator 2a, which then gives the quadratic formula:
x = [-b + sqr(b^2 -4ac)]/2a
x = minus b plus or minus the square root of (b^2-4ac), all divided by 2a
where a is the coefficient of the x^2 term, b is the coefficient of the x term and c = the constant term
b^2-4ac is called the discriminant that tells you if there are 2 real roots, 2 imaginary roots or 2 repeating roots. if b^2-4ac>0 there's 2 real, if <0 then 2 imaginary and if=0 then 2 repeating roots. 2 real roots or zeroes are where the graph intersects the x axis. 2 imaginary roots mean there is no intersection with the x axis. 2 repeating roots mean the graph just touches the x axis.
the -b/2a part is the line of symmetry, x=-b/2a for the parabola graph
