Solve 7x+2x=85x+x^2+2

$$7x+2x = 85x+ { x }^{ 2 } +2$$

Solve for x (complex solution)

$$x=\sqrt{1442}-38\approx -0.026324908$$

$$x=-\left(\sqrt{1442}+38\right)\approx -75.973675092$$

Steps Using the Quadratic Formula

Combine 7x and 2x to get 9x.

$$9x=85x+x^{2}+2$$

Subtract 85x from both sides.

$$9x-85x=x^{2}+2$$

Combine 9x and -85x to get -76x.

$$-76x=x^{2}+2$$

Subtract x^{2} from both sides.

$$-76x-x^{2}=2$$

Subtract 2 from both sides.

$$-76x-x^{2}-2=0$$

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.

$$-x^{2}-76x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -76 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Square -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-76\right)±\sqrt{5776+4\left(-2\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.

$$x=\frac{-\left(-76\right)±\sqrt{5776-8}}{2\left(-1\right)}$$

Add 5776 to -8.

$$x=\frac{-\left(-76\right)±\sqrt{5768}}{2\left(-1\right)}$$

Take the square root of 5768.

$$x=\frac{-\left(-76\right)±2\sqrt{1442}}{2\left(-1\right)}$$

The opposite of -76 is 76.

$$x=\frac{76±2\sqrt{1442}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{76±2\sqrt{1442}}{-2}$$

Now solve the equation x=\frac{76±2\sqrt{1442}}{-2} when ± is plus. Add 76 to 2\sqrt{1442}.

$$x=\frac{2\sqrt{1442}+76}{-2}$$

Divide 76+2\sqrt{1442} by -2.

$$x=-\left(\sqrt{1442}+38\right)$$

Now solve the equation x=\frac{76±2\sqrt{1442}}{-2} when ± is minus. Subtract 2\sqrt{1442} from 76.

$$x=\frac{76-2\sqrt{1442}}{-2}$$

Divide 76-2\sqrt{1442} by -2.

$$x=\sqrt{1442}-38$$

The equation is now solved.

$$x=-\left(\sqrt{1442}+38\right) x=\sqrt{1442}-38$$

Steps for Completing the Square

Combine 7x and 2x to get 9x.

$$9x=85x+x^{2}+2$$

Subtract 85x from both sides.

$$9x-85x=x^{2}+2$$

Combine 9x and -85x to get -76x.

$$-76x=x^{2}+2$$

Subtract x^{2} from both sides.

$$-76x-x^{2}=2$$

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.

$$-x^{2}-76x=2$$

Divide both sides by -1.

$$\frac{-x^{2}-76x}{-1}=\frac{2}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{76}{-1}\right)x=\frac{2}{-1}$$

Divide -76 by -1.

$$x^{2}+76x=\frac{2}{-1}$$

Divide 2 by -1.

$$x^{2}+76x=-2$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=-2+38^{2}$$

Square 38.

$$x^{2}+76x+1444=-2+1444$$

Add -2 to 1444.

$$x^{2}+76x+1444=1442$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=1442$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{1442}$$

Simplify.

$$x+38=\sqrt{1442} x+38=-\sqrt{1442}$$

Subtract 38 from both sides of the equation.

$$x=\sqrt{1442}-38 x=-\sqrt{1442}-38$$

Solve for x

$$x=\sqrt{1442}-38\approx -0.026324908$$

$$x=-\sqrt{1442}-38\approx -75.973675092$$

Steps Using the Quadratic Formula

Combine 7x and 2x to get 9x.

$$9x=85x+x^{2}+2$$

Subtract 85x from both sides.

$$9x-85x=x^{2}+2$$

Combine 9x and -85x to get -76x.

$$-76x=x^{2}+2$$

Subtract x^{2} from both sides.

$$-76x-x^{2}=2$$

Subtract 2 from both sides.

$$-76x-x^{2}-2=0$$

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.

$$-x^{2}-76x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -76 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Square -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-76\right)±\sqrt{5776+4\left(-2\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.

$$x=\frac{-\left(-76\right)±\sqrt{5776-8}}{2\left(-1\right)}$$

Add 5776 to -8.

$$x=\frac{-\left(-76\right)±\sqrt{5768}}{2\left(-1\right)}$$

Take the square root of 5768.

$$x=\frac{-\left(-76\right)±2\sqrt{1442}}{2\left(-1\right)}$$

The opposite of -76 is 76.

$$x=\frac{76±2\sqrt{1442}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{76±2\sqrt{1442}}{-2}$$

Now solve the equation x=\frac{76±2\sqrt{1442}}{-2} when ± is plus. Add 76 to 2\sqrt{1442}.

$$x=\frac{2\sqrt{1442}+76}{-2}$$

Divide 76+2\sqrt{1442} by -2.

$$x=-\left(\sqrt{1442}+38\right)$$

Now solve the equation x=\frac{76±2\sqrt{1442}}{-2} when ± is minus. Subtract 2\sqrt{1442} from 76.

$$x=\frac{76-2\sqrt{1442}}{-2}$$

Divide 76-2\sqrt{1442} by -2.

$$x=\sqrt{1442}-38$$

The equation is now solved.

$$x=-\left(\sqrt{1442}+38\right) x=\sqrt{1442}-38$$

Steps for Completing the Square

Combine 7x and 2x to get 9x.

$$9x=85x+x^{2}+2$$

Subtract 85x from both sides.

$$9x-85x=x^{2}+2$$

Combine 9x and -85x to get -76x.

$$-76x=x^{2}+2$$

Subtract x^{2} from both sides.

$$-76x-x^{2}=2$$

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.

$$-x^{2}-76x=2$$

Divide both sides by -1.

$$\frac{-x^{2}-76x}{-1}=\frac{2}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{76}{-1}\right)x=\frac{2}{-1}$$

Divide -76 by -1.

$$x^{2}+76x=\frac{2}{-1}$$

Divide 2 by -1.

$$x^{2}+76x=-2$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=-2+38^{2}$$

Square 38.

$$x^{2}+76x+1444=-2+1444$$

Add -2 to 1444.

$$x^{2}+76x+1444=1442$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=1442$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{1442}$$

Simplify.

$$x+38=\sqrt{1442} x+38=-\sqrt{1442}$$

Subtract 38 from both sides of the equation.

$$x=\sqrt{1442}-38 x=-\sqrt{1442}-38$$