Solve xleft(x+1right)/2=1923(x-1)

$$\frac{ x \left( x+1 \right) }{ 2 } = 1923(x-1)$$

Solve for x

$$x = \frac{\sqrt{14768641} + 3845}{2} \approx 3843.999479573$$

$$x = \frac{3845 - \sqrt{14768641}}{2} \approx 1.000520427$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(x+1\right)=3846\left(x-1\right)$$

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

$$x^{2}+x=3846\left(x-1\right)$$

Use the distributive property to multiply 3846 by x-1.

$$x^{2}+x=3846x-3846$$

Subtract 3846x from both sides.

$$x^{2}+x-3846x=-3846$$

Combine x and -3846x to get -3845x.

$$x^{2}-3845x=-3846$$

Add 3846 to both sides.

$$x^{2}-3845x+3846=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3845 for b, and 3846 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3845\right)±\sqrt{\left(-3845\right)^{2}-4\times 3846}}{2}$$

Square -3845.

$$x=\frac{-\left(-3845\right)±\sqrt{14784025-4\times 3846}}{2}$$

Multiply -4 times 3846.

$$x=\frac{-\left(-3845\right)±\sqrt{14784025-15384}}{2}$$

Add 14784025 to -15384.

$$x=\frac{-\left(-3845\right)±\sqrt{14768641}}{2}$$

The opposite of -3845 is 3845.

$$x=\frac{3845±\sqrt{14768641}}{2}$$

Now solve the equation x=\frac{3845±\sqrt{14768641}}{2} when ± is plus. Add 3845 to \sqrt{14768641}.

$$x=\frac{\sqrt{14768641}+3845}{2}$$

Now solve the equation x=\frac{3845±\sqrt{14768641}}{2} when ± is minus. Subtract \sqrt{14768641} from 3845.

$$x=\frac{3845-\sqrt{14768641}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{14768641}+3845}{2} x=\frac{3845-\sqrt{14768641}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(x+1\right)=3846\left(x-1\right)$$

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

$$x^{2}+x=3846\left(x-1\right)$$

Use the distributive property to multiply 3846 by x-1.

$$x^{2}+x=3846x-3846$$

Subtract 3846x from both sides.

$$x^{2}+x-3846x=-3846$$

Combine x and -3846x to get -3845x.

$$x^{2}-3845x=-3846$$

Divide -3845, the coefficient of the x term, by 2 to get -\frac{3845}{2}. Then add the square of -\frac{3845}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3845x+\left(-\frac{3845}{2}\right)^{2}=-3846+\left(-\frac{3845}{2}\right)^{2}$$

Square -\frac{3845}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3845x+\frac{14784025}{4}=-3846+\frac{14784025}{4}$$

Add -3846 to \frac{14784025}{4}.

$$x^{2}-3845x+\frac{14784025}{4}=\frac{14768641}{4}$$

Factor x^{2}-3845x+\frac{14784025}{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}.

$$\left(x-\frac{3845}{2}\right)^{2}=\frac{14768641}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3845}{2}\right)^{2}}=\sqrt{\frac{14768641}{4}}$$

Simplify.

$$x-\frac{3845}{2}=\frac{\sqrt{14768641}}{2} x-\frac{3845}{2}=-\frac{\sqrt{14768641}}{2}$$

Add \frac{3845}{2} to both sides of the equation.

$$x=\frac{\sqrt{14768641}+3845}{2} x=\frac{3845-\sqrt{14768641}}{2}$$