Solve x^2-66x+97=0

$${ x }^{ 2 } -66x+97=0$$

Solve for x

$$x=4\sqrt{62}+33\approx 64.496031496$$

$$x=33-4\sqrt{62}\approx 1.503968504$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-66x+97=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -66 for b, and 97 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 97}}{2}$$

Square -66.

$$x=\frac{-\left(-66\right)±\sqrt{4356-4\times 97}}{2}$$

Multiply -4 times 97.

$$x=\frac{-\left(-66\right)±\sqrt{4356-388}}{2}$$

Add 4356 to -388.

$$x=\frac{-\left(-66\right)±\sqrt{3968}}{2}$$

Take the square root of 3968.

$$x=\frac{-\left(-66\right)±8\sqrt{62}}{2}$$

The opposite of -66 is 66.

$$x=\frac{66±8\sqrt{62}}{2}$$

Now solve the equation x=\frac{66±8\sqrt{62}}{2} when ± is plus. Add 66 to 8\sqrt{62}.

$$x=\frac{8\sqrt{62}+66}{2}$$

Divide 66+8\sqrt{62} by 2.

$$x=4\sqrt{62}+33$$

Now solve the equation x=\frac{66±8\sqrt{62}}{2} when ± is minus. Subtract 8\sqrt{62} from 66.

$$x=\frac{66-8\sqrt{62}}{2}$$

Divide 66-8\sqrt{62} by 2.

$$x=33-4\sqrt{62}$$

The equation is now solved.

$$x=4\sqrt{62}+33 x=33-4\sqrt{62}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-66x+97=0$$

Subtract 97 from both sides of the equation.

$$x^{2}-66x+97-97=-97$$

Subtracting 97 from itself leaves 0.

$$x^{2}-66x=-97$$

Divide -66, the coefficient of the x term, by 2 to get -33. Then add the square of -33 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-66x+\left(-33\right)^{2}=-97+\left(-33\right)^{2}$$

Square -33.

$$x^{2}-66x+1089=-97+1089$$

Add -97 to 1089.

$$x^{2}-66x+1089=992$$

Factor x^{2}-66x+1089. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-33\right)^{2}=992$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-33\right)^{2}}=\sqrt{992}$$

Simplify.

$$x-33=4\sqrt{62} x-33=-4\sqrt{62}$$

Add 33 to both sides of the equation.

$$x=4\sqrt{62}+33 x=33-4\sqrt{62}$$