Solve (x-2)*(x-2)=103635

$$( x - 2 ) \times ( x - 2 ) = 103635$$

Solve for x

$$x=21\sqrt{235}+2\approx 323.923904052$$

$$x=2-21\sqrt{235}\approx -319.923904052$$

Steps Using the Quadratic Formula

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

$$\left(x-2\right)^{2}=103635$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4=103635$$

Subtract 103635 from both sides.

$$x^{2}-4x+4-103635=0$$

Subtract 103635 from 4 to get -103631.

$$x^{2}-4x-103631=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -103631 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-103631\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-103631\right)}}{2}$$

Multiply -4 times -103631.

$$x=\frac{-\left(-4\right)±\sqrt{16+414524}}{2}$$

Add 16 to 414524.

$$x=\frac{-\left(-4\right)±\sqrt{414540}}{2}$$

Take the square root of 414540.

$$x=\frac{-\left(-4\right)±42\sqrt{235}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±42\sqrt{235}}{2}$$

Now solve the equation x=\frac{4±42\sqrt{235}}{2} when ± is plus. Add 4 to 42\sqrt{235}.

$$x=\frac{42\sqrt{235}+4}{2}$$

Divide 4+42\sqrt{235} by 2.

$$x=21\sqrt{235}+2$$

Now solve the equation x=\frac{4±42\sqrt{235}}{2} when ± is minus. Subtract 42\sqrt{235} from 4.

$$x=\frac{4-42\sqrt{235}}{2}$$

Divide 4-42\sqrt{235} by 2.

$$x=2-21\sqrt{235}$$

The equation is now solved.

$$x=21\sqrt{235}+2 x=2-21\sqrt{235}$$

Steps for Completing the Square

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

$$\left(x-2\right)^{2}=103635$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{103635}$$

Simplify.

$$x-2=21\sqrt{235} x-2=-21\sqrt{235}$$

Add 2 to both sides of the equation.

$$x=21\sqrt{235}+2 x=2-21\sqrt{235}$$