A quadratic function in standard form (y = ax^2 + bx + cy=ax2+bx+c) can be solved using its related quadratic equation (ax^2 + bx + c = 0ax2+bx+c=0) and the quadratic formula. The quadratic formula is:
x = (-b +-sqrt(b^2 - 4ac))/(2a)x=−b±√b2−4ac2a
For the quadratic function y = -x^2 - 6x - 6y=−x2−6x−6, a = -1a=−1, b = -6b=−6, and c = -6c=−6. Substitute these values into the quadratic formula and simplify following the Order of Operations.
x = (-(-6) +- sqrt((-6)^2 - 4(-1)(-6)))/((2)(-1))x=−(−6)±√(−6)2−4(−1)(−6)(2)(−1)
x = (6 +- sqrt(36 - 24))/-2x=6±√36−24−2
x = (6 +- sqrt(12))/-2x=6±√12−2
x = (-6 +- 2sqrt3)/-2x=−6±2√3−2
x = 3 +- sqrt3x=3±√3
So, the roots of y = -x ^2 -6x -6y=−x2−6x−6 are x = 3 + sqrt3x=3+√3 and x = 3 - sqrt3x=3−√3.