Solve x(x+1)/2=646

$$\frac{ x(x+1) }{ 2 } =646$$

Solve for x

$$x = \frac{\sqrt{5169} - 1}{2} \approx 35.447878936$$

$$x=\frac{-\sqrt{5169}-1}{2}\approx -36.447878936$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$x\left(x+1\right)=646\times 2$$

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

$$x^{2}+x=646\times 2$$

Multiply 646 and 2 to get 1292.

$$x^{2}+x=1292$$

Subtract 1292 from both sides.

$$x^{2}+x-1292=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -1292 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1292\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-1292\right)}}{2}$$

Multiply -4 times -1292.

$$x=\frac{-1±\sqrt{1+5168}}{2}$$

Add 1 to 5168.

$$x=\frac{-1±\sqrt{5169}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{5169}}{2} when ± is plus. Add -1 to \sqrt{5169}.

$$x=\frac{\sqrt{5169}-1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{5169}}{2} when ± is minus. Subtract \sqrt{5169} from -1.

$$x=\frac{-\sqrt{5169}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{5169}-1}{2} x=\frac{-\sqrt{5169}-1}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$x\left(x+1\right)=646\times 2$$

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

$$x^{2}+x=646\times 2$$

Multiply 646 and 2 to get 1292.

$$x^{2}+x=1292$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=1292+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=1292+\frac{1}{4}$$

Add 1292 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{5169}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{5169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{5169}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{5169}}{2} x+\frac{1}{2}=-\frac{\sqrt{5169}}{2}$$

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

$$x=\frac{\sqrt{5169}-1}{2} x=\frac{-\sqrt{5169}-1}{2}$$