Solve (1+x-1)*(x-1)div2+x=153

$$(1+x-1) \times (x-1) \div 2+x=153$$

Solve for x

$$x=-18$$

$$x=17$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$\left(1+x-1\right)\left(x-1\right)+2x=306$$

Subtract 1 from 1 to get 0.

$$x\left(x-1\right)+2x=306$$

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

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

Combine -x and 2x to get x.

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

Subtract 306 from both sides.

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

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}.

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

Square 1.

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

Multiply -4 times -306.

$$x=\frac{-1±\sqrt{1+1224}}{2}$$

Add 1 to 1224.

$$x=\frac{-1±\sqrt{1225}}{2}$$

Take the square root of 1225.

$$x=\frac{-1±35}{2}$$

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

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

Divide 34 by 2.

$$x=17$$

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

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

Divide -36 by 2.

$$x=-18$$

The equation is now solved.

$$x=17 x=-18$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$\left(1+x-1\right)\left(x-1\right)+2x=306$$

Subtract 1 from 1 to get 0.

$$x\left(x-1\right)+2x=306$$

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

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

Combine -x and 2x to get x.

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

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.

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

Subtract \frac{1}{2} from both sides of the equation.

$$x=17 x=-18$$