Solve 123*123=x+xx

$$123 \times 123=x+xx$$

Solve for x

$$x = \frac{\sqrt{60517} - 1}{2} \approx 122.501016256$$

$$x=\frac{-\sqrt{60517}-1}{2}\approx -123.501016256$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$123\times 123=x+x^{2}$$

Multiply 123 and 123 to get 15129.

$$15129=x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x+x^{2}=15129$$

Subtract 15129 from both sides.

$$x+x^{2}-15129=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+x-15129=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -15129 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-15129\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-15129\right)}}{2}$$

Multiply -4 times -15129.

$$x=\frac{-1±\sqrt{1+60516}}{2}$$

Add 1 to 60516.

$$x=\frac{-1±\sqrt{60517}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{60517}}{2} when ± is plus. Add -1 to \sqrt{60517}.

$$x=\frac{\sqrt{60517}-1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{60517}}{2} when ± is minus. Subtract \sqrt{60517} from -1.

$$x=\frac{-\sqrt{60517}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{60517}-1}{2} x=\frac{-\sqrt{60517}-1}{2}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$123\times 123=x+x^{2}$$

Multiply 123 and 123 to get 15129.

$$15129=x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x+x^{2}=15129$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+x=15129$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=15129+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=15129+\frac{1}{4}$$

Add 15129 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{60517}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{60517}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{60517}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{60517}}{2} x+\frac{1}{2}=-\frac{\sqrt{60517}}{2}$$

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

$$x=\frac{\sqrt{60517}-1}{2} x=\frac{-\sqrt{60517}-1}{2}$$