Solve x=8903quad1+2+3+4+5+6/x-1263

$$x = 8903 \quad \frac { 1 + 2 + 3 + 4 + 5 + 6 } { x - 1263 }$$

Solve for x

$$x = \frac{\sqrt{2343021} + 1263}{2} \approx 1396.84649016$$

$$x=\frac{1263-\sqrt{2343021}}{2}\approx -133.84649016$$

Steps Using the Quadratic Formula

Add 1 and 2 to get 3.

$$x=8903\times \frac{3+3+4+5+6}{x-1263}$$

Add 3 and 3 to get 6.

$$x=8903\times \frac{6+4+5+6}{x-1263}$$

Add 6 and 4 to get 10.

$$x=8903\times \frac{10+5+6}{x-1263}$$

Add 10 and 5 to get 15.

$$x=8903\times \frac{15+6}{x-1263}$$

Add 15 and 6 to get 21.

$$x=8903\times \frac{21}{x-1263}$$

Express 8903\times \frac{21}{x-1263} as a single fraction.

$$x=\frac{8903\times 21}{x-1263}$$

Multiply 8903 and 21 to get 186963.

$$x=\frac{186963}{x-1263}$$

Subtract \frac{186963}{x-1263} from both sides.

$$x-\frac{186963}{x-1263}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-1263}{x-1263}.

$$\frac{x\left(x-1263\right)}{x-1263}-\frac{186963}{x-1263}=0$$

Since \frac{x\left(x-1263\right)}{x-1263} and \frac{186963}{x-1263} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-1263\right)-186963}{x-1263}=0$$

Do the multiplications in x\left(x-1263\right)-186963.

$$\frac{x^{2}-1263x-186963}{x-1263}=0$$

Variable x cannot be equal to 1263 since division by zero is not defined. Multiply both sides of the equation by x-1263.

$$x^{2}-1263x-186963=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1263 for b, and -186963 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1263\right)±\sqrt{\left(-1263\right)^{2}-4\left(-186963\right)}}{2}$$

Square -1263.

$$x=\frac{-\left(-1263\right)±\sqrt{1595169-4\left(-186963\right)}}{2}$$

Multiply -4 times -186963.

$$x=\frac{-\left(-1263\right)±\sqrt{1595169+747852}}{2}$$

Add 1595169 to 747852.

$$x=\frac{-\left(-1263\right)±\sqrt{2343021}}{2}$$

The opposite of -1263 is 1263.

$$x=\frac{1263±\sqrt{2343021}}{2}$$

Now solve the equation x=\frac{1263±\sqrt{2343021}}{2} when ± is plus. Add 1263 to \sqrt{2343021}.

$$x=\frac{\sqrt{2343021}+1263}{2}$$

Now solve the equation x=\frac{1263±\sqrt{2343021}}{2} when ± is minus. Subtract \sqrt{2343021} from 1263.

$$x=\frac{1263-\sqrt{2343021}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2343021}+1263}{2} x=\frac{1263-\sqrt{2343021}}{2}$$

Steps for Completing the Square

Add 1 and 2 to get 3.

$$x=8903\times \frac{3+3+4+5+6}{x-1263}$$

Add 3 and 3 to get 6.

$$x=8903\times \frac{6+4+5+6}{x-1263}$$

Add 6 and 4 to get 10.

$$x=8903\times \frac{10+5+6}{x-1263}$$

Add 10 and 5 to get 15.

$$x=8903\times \frac{15+6}{x-1263}$$

Add 15 and 6 to get 21.

$$x=8903\times \frac{21}{x-1263}$$

Express 8903\times \frac{21}{x-1263} as a single fraction.

$$x=\frac{8903\times 21}{x-1263}$$

Multiply 8903 and 21 to get 186963.

$$x=\frac{186963}{x-1263}$$

Subtract \frac{186963}{x-1263} from both sides.

$$x-\frac{186963}{x-1263}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-1263}{x-1263}.

$$\frac{x\left(x-1263\right)}{x-1263}-\frac{186963}{x-1263}=0$$

Since \frac{x\left(x-1263\right)}{x-1263} and \frac{186963}{x-1263} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-1263\right)-186963}{x-1263}=0$$

Do the multiplications in x\left(x-1263\right)-186963.

$$\frac{x^{2}-1263x-186963}{x-1263}=0$$

Variable x cannot be equal to 1263 since division by zero is not defined. Multiply both sides of the equation by x-1263.

$$x^{2}-1263x-186963=0$$

Add 186963 to both sides. Anything plus zero gives itself.

$$x^{2}-1263x=186963$$

Divide -1263, the coefficient of the x term, by 2 to get -\frac{1263}{2}. Then add the square of -\frac{1263}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1263x+\left(-\frac{1263}{2}\right)^{2}=186963+\left(-\frac{1263}{2}\right)^{2}$$

Square -\frac{1263}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1263x+\frac{1595169}{4}=186963+\frac{1595169}{4}$$

Add 186963 to \frac{1595169}{4}.

$$x^{2}-1263x+\frac{1595169}{4}=\frac{2343021}{4}$$

Factor x^{2}-1263x+\frac{1595169}{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}.

$$\left(x-\frac{1263}{2}\right)^{2}=\frac{2343021}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1263}{2}\right)^{2}}=\sqrt{\frac{2343021}{4}}$$

Simplify.

$$x-\frac{1263}{2}=\frac{\sqrt{2343021}}{2} x-\frac{1263}{2}=-\frac{\sqrt{2343021}}{2}$$

Add \frac{1263}{2} to both sides of the equation.

$$x=\frac{\sqrt{2343021}+1263}{2} x=\frac{1263-\sqrt{2343021}}{2}$$