Solve 6(135)={left(x-2*1/2right)}^2

$$6(135)= { \left(x-2 \times \frac{ 1 }{ 2 } \right) }^{ 2 }$$

Solve for x

$$x=9\sqrt{10}+1\approx 29.460498942$$

$$x=1-9\sqrt{10}\approx -27.460498942$$

Steps Using the Quadratic Formula

Multiply 6 and 135 to get 810.

$$810=\left(x-2\times \frac{1}{2}\right)^{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$810=\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$810=x^{2}-2x+1$$

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

$$x^{2}-2x+1=810$$

Subtract 810 from both sides.

$$x^{2}-2x+1-810=0$$

Subtract 810 from 1 to get -809.

$$x^{2}-2x-809=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -809 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-809\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-809\right)}}{2}$$

Multiply -4 times -809.

$$x=\frac{-\left(-2\right)±\sqrt{4+3236}}{2}$$

Add 4 to 3236.

$$x=\frac{-\left(-2\right)±\sqrt{3240}}{2}$$

Take the square root of 3240.

$$x=\frac{-\left(-2\right)±18\sqrt{10}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±18\sqrt{10}}{2}$$

Now solve the equation x=\frac{2±18\sqrt{10}}{2} when ± is plus. Add 2 to 18\sqrt{10}.

$$x=\frac{18\sqrt{10}+2}{2}$$

Divide 2+18\sqrt{10} by 2.

$$x=9\sqrt{10}+1$$

Now solve the equation x=\frac{2±18\sqrt{10}}{2} when ± is minus. Subtract 18\sqrt{10} from 2.

$$x=\frac{2-18\sqrt{10}}{2}$$

Divide 2-18\sqrt{10} by 2.

$$x=1-9\sqrt{10}$$

The equation is now solved.

$$x=9\sqrt{10}+1 x=1-9\sqrt{10}$$

Steps for Completing the Square

Multiply 6 and 135 to get 810.

$$810=\left(x-2\times \frac{1}{2}\right)^{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$810=\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$810=x^{2}-2x+1$$

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

$$x^{2}-2x+1=810$$

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

$$\left(x-1\right)^{2}=810$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{810}$$

Simplify.

$$x-1=9\sqrt{10} x-1=-9\sqrt{10}$$

Add 1 to both sides of the equation.

$$x=9\sqrt{10}+1 x=1-9\sqrt{10}$$