Solve x(x+1)-(x+(x+1))=305

$$x(x+1)-(x+(x+1))=305$$

Solve for x

$$x=-17$$

$$x=18$$

Steps Using the Quadratic Formula

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

$$x^{2}+x-\left(x+x+1\right)=305$$

Combine x and x to get 2x.

$$x^{2}+x-\left(2x+1\right)=305$$

To find the opposite of 2x+1, find the opposite of each term.

$$x^{2}+x-2x-1=305$$

Combine x and -2x to get -x.

$$x^{2}-x-1=305$$

Subtract 305 from both sides.

$$x^{2}-x-1-305=0$$

Subtract 305 from -1 to get -306.

$$x^{2}-x-306=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -306 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-306\right)}}{2}$$

Multiply -4 times -306.

$$x=\frac{-\left(-1\right)±\sqrt{1+1224}}{2}$$

Add 1 to 1224.

$$x=\frac{-\left(-1\right)±\sqrt{1225}}{2}$$

Take the square root of 1225.

$$x=\frac{-\left(-1\right)±35}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±35}{2}$$

Now solve the equation x=\frac{1±35}{2} when ± is plus. Add 1 to 35.

$$x=\frac{36}{2}$$

Divide 36 by 2.

$$x=18$$

Now solve the equation x=\frac{1±35}{2} when ± is minus. Subtract 35 from 1.

$$x=-\frac{34}{2}$$

Divide -34 by 2.

$$x=-17$$

The equation is now solved.

$$x=18 x=-17$$

Steps for Completing the Square

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

$$x^{2}+x-\left(x+x+1\right)=305$$

Combine x and x to get 2x.

$$x^{2}+x-\left(2x+1\right)=305$$

To find the opposite of 2x+1, find the opposite of each term.

$$x^{2}+x-2x-1=305$$

Combine x and -2x to get -x.

$$x^{2}-x-1=305$$

Add 1 to both sides.

$$x^{2}-x=305+1$$

Add 305 and 1 to get 306.

$$x^{2}-x=306$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=306+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=306+\frac{1}{4}$$

Add 306 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{1225}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{1225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{35}{2} x-\frac{1}{2}=-\frac{35}{2}$$

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

$$x=18 x=-17$$