Solve 1035=x(x-1)

$$1035=x(x-1)$$

Solve for x

$$x = \frac{\sqrt{4141} + 1}{2} \approx 32.675301086$$

$$x=\frac{1-\sqrt{4141}}{2}\approx -31.675301086$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x-1.

$$1035=x^{2}-x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=1035$$

Subtract 1035 from both sides.

$$x^{2}-x-1035=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -1035 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1035\right)}}{2}$$

Multiply -4 times -1035.

$$x=\frac{-\left(-1\right)±\sqrt{1+4140}}{2}$$

Add 1 to 4140.

$$x=\frac{-\left(-1\right)±\sqrt{4141}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{4141}}{2}$$

Now solve the equation x=\frac{1±\sqrt{4141}}{2} when ± is plus. Add 1 to \sqrt{4141}.

$$x=\frac{\sqrt{4141}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{4141}}{2} when ± is minus. Subtract \sqrt{4141} from 1.

$$x=\frac{1-\sqrt{4141}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{4141}+1}{2} x=\frac{1-\sqrt{4141}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x by x-1.

$$1035=x^{2}-x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=1035$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=1035+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=1035+\frac{1}{4}$$

Add 1035 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{4141}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{4141}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{4141}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{4141}}{2} x-\frac{1}{2}=-\frac{\sqrt{4141}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{4141}+1}{2} x=\frac{1-\sqrt{4141}}{2}$$