Solve x+120*66/1266-x=76

$$x + \frac { 120 \times 66 } { 1266 - x } = 76$$

Solve for x

$$x=\sqrt{361945}+671\approx 1272.618649977$$

$$x=671-\sqrt{361945}\approx 69.381350023$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1266 since division by zero is not defined. Multiply both sides of the equation by -x+1266.

$$\left(-x+1266\right)x+120\times 66=76\left(-x+1266\right)$$

Use the distributive property to multiply -x+1266 by x.

$$-x^{2}+1266x+120\times 66=76\left(-x+1266\right)$$

Multiply 120 and 66 to get 7920.

$$-x^{2}+1266x+7920=76\left(-x+1266\right)$$

Use the distributive property to multiply 76 by -x+1266.

$$-x^{2}+1266x+7920=-76x+96216$$

Add 76x to both sides.

$$-x^{2}+1266x+7920+76x=96216$$

Combine 1266x and 76x to get 1342x.

$$-x^{2}+1342x+7920=96216$$

Subtract 96216 from both sides.

$$-x^{2}+1342x+7920-96216=0$$

Subtract 96216 from 7920 to get -88296.

$$-x^{2}+1342x-88296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1342 for b, and -88296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1342±\sqrt{1342^{2}-4\left(-1\right)\left(-88296\right)}}{2\left(-1\right)}$$

Square 1342.

$$x=\frac{-1342±\sqrt{1800964-4\left(-1\right)\left(-88296\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1342±\sqrt{1800964+4\left(-88296\right)}}{2\left(-1\right)}$$

Multiply 4 times -88296.

$$x=\frac{-1342±\sqrt{1800964-353184}}{2\left(-1\right)}$$

Add 1800964 to -353184.

$$x=\frac{-1342±\sqrt{1447780}}{2\left(-1\right)}$$

Take the square root of 1447780.

$$x=\frac{-1342±2\sqrt{361945}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1342±2\sqrt{361945}}{-2}$$

Now solve the equation x=\frac{-1342±2\sqrt{361945}}{-2} when ± is plus. Add -1342 to 2\sqrt{361945}.

$$x=\frac{2\sqrt{361945}-1342}{-2}$$

Divide -1342+2\sqrt{361945} by -2.

$$x=671-\sqrt{361945}$$

Now solve the equation x=\frac{-1342±2\sqrt{361945}}{-2} when ± is minus. Subtract 2\sqrt{361945} from -1342.

$$x=\frac{-2\sqrt{361945}-1342}{-2}$$

Divide -1342-2\sqrt{361945} by -2.

$$x=\sqrt{361945}+671$$

The equation is now solved.

$$x=671-\sqrt{361945} x=\sqrt{361945}+671$$

Steps for Completing the Square

Variable x cannot be equal to 1266 since division by zero is not defined. Multiply both sides of the equation by -x+1266.

$$\left(-x+1266\right)x+120\times 66=76\left(-x+1266\right)$$

Use the distributive property to multiply -x+1266 by x.

$$-x^{2}+1266x+120\times 66=76\left(-x+1266\right)$$

Multiply 120 and 66 to get 7920.

$$-x^{2}+1266x+7920=76\left(-x+1266\right)$$

Use the distributive property to multiply 76 by -x+1266.

$$-x^{2}+1266x+7920=-76x+96216$$

Add 76x to both sides.

$$-x^{2}+1266x+7920+76x=96216$$

Combine 1266x and 76x to get 1342x.

$$-x^{2}+1342x+7920=96216$$

Subtract 7920 from both sides.

$$-x^{2}+1342x=96216-7920$$

Subtract 7920 from 96216 to get 88296.

$$-x^{2}+1342x=88296$$

Divide both sides by -1.

$$\frac{-x^{2}+1342x}{-1}=\frac{88296}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1342}{-1}x=\frac{88296}{-1}$$

Divide 1342 by -1.

$$x^{2}-1342x=\frac{88296}{-1}$$

Divide 88296 by -1.

$$x^{2}-1342x=-88296$$

Divide -1342, the coefficient of the x term, by 2 to get -671. Then add the square of -671 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1342x+\left(-671\right)^{2}=-88296+\left(-671\right)^{2}$$

Square -671.

$$x^{2}-1342x+450241=-88296+450241$$

Add -88296 to 450241.

$$x^{2}-1342x+450241=361945$$

Factor x^{2}-1342x+450241. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-671\right)^{2}=361945$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-671\right)^{2}}=\sqrt{361945}$$

Simplify.

$$x-671=\sqrt{361945} x-671=-\sqrt{361945}$$

Add 671 to both sides of the equation.

$$x=\sqrt{361945}+671 x=671-\sqrt{361945}$$