Solve (x+x+1-4)(x-0.5)/2=405

$$\frac { ( x + x + 1 - 4 ) ( x - 0.5 ) } { 2 } = 405$$

Solve for x

$$x=\frac{\sqrt{1621}}{2}+1\approx 21.130822139$$

$$x=-\frac{\sqrt{1621}}{2}+1\approx -19.130822139$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$\left(x+x+1-4\right)\left(x-0.5\right)=405\times 2$$

Combine x and x to get 2x.

$$\left(2x+1-4\right)\left(x-0.5\right)=405\times 2$$

Subtract 4 from 1 to get -3.

$$\left(2x-3\right)\left(x-0.5\right)=405\times 2$$

Use the distributive property to multiply 2x-3 by x-0.5 and combine like terms.

$$2x^{2}-4x+1.5=405\times 2$$

Multiply 405 and 2 to get 810.

$$2x^{2}-4x+1.5=810$$

Subtract 810 from both sides.

$$2x^{2}-4x+1.5-810=0$$

Subtract 810 from 1.5 to get -808.5.

$$2x^{2}-4x-808.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -808.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-808.5\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-808.5\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-808.5\right)}}{2\times 2}$$

Multiply -8 times -808.5.

$$x=\frac{-\left(-4\right)±\sqrt{16+6468}}{2\times 2}$$

Add 16 to 6468.

$$x=\frac{-\left(-4\right)±\sqrt{6484}}{2\times 2}$$

Take the square root of 6484.

$$x=\frac{-\left(-4\right)±2\sqrt{1621}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{1621}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±2\sqrt{1621}}{4}$$

Now solve the equation x=\frac{4±2\sqrt{1621}}{4} when ± is plus. Add 4 to 2\sqrt{1621}.

$$x=\frac{2\sqrt{1621}+4}{4}$$

Divide 4+2\sqrt{1621} by 4.

$$x=\frac{\sqrt{1621}}{2}+1$$

Now solve the equation x=\frac{4±2\sqrt{1621}}{4} when ± is minus. Subtract 2\sqrt{1621} from 4.

$$x=\frac{4-2\sqrt{1621}}{4}$$

Divide 4-2\sqrt{1621} by 4.

$$x=-\frac{\sqrt{1621}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{1621}}{2}+1 x=-\frac{\sqrt{1621}}{2}+1$$

Steps for Completing the Square

Multiply both sides by 2.

$$\left(x+x+1-4\right)\left(x-0.5\right)=405\times 2$$

Combine x and x to get 2x.

$$\left(2x+1-4\right)\left(x-0.5\right)=405\times 2$$

Subtract 4 from 1 to get -3.

$$\left(2x-3\right)\left(x-0.5\right)=405\times 2$$

Use the distributive property to multiply 2x-3 by x-0.5 and combine like terms.

$$2x^{2}-4x+1.5=405\times 2$$

Multiply 405 and 2 to get 810.

$$2x^{2}-4x+1.5=810$$

Subtract 1.5 from both sides.

$$2x^{2}-4x=810-1.5$$

Subtract 1.5 from 810 to get 808.5.

$$2x^{2}-4x=808.5$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{808.5}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{808.5}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{808.5}{2}$$

Divide 808.5 by 2.

$$x^{2}-2x=404.25$$

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

$$x^{2}-2x+1=404.25+1$$

Add 404.25 to 1.

$$x^{2}-2x+1=405.25$$

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}.

$$\left(x-1\right)^{2}=405.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{405.25}$$

Simplify.

$$x-1=\frac{\sqrt{1621}}{2} x-1=-\frac{\sqrt{1621}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{1621}}{2}+1 x=-\frac{\sqrt{1621}}{2}+1$$