Solve 14-7x+20+25+22129+x/x=x

$$\frac { 14 - 7 x + 20 + 25 + 22129 + x } { x } = x$$

Solve for x

$$x=7\sqrt{453}-3\approx 145.986576577$$

$$x=-7\sqrt{453}-3\approx -151.986576577$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$14-7x+20+25+22129+x=xx$$

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

$$14-7x+20+25+22129+x=x^{2}$$

Add 14 and 20 to get 34.

$$34-7x+25+22129+x=x^{2}$$

Add 34 and 25 to get 59.

$$59-7x+22129+x=x^{2}$$

Add 59 and 22129 to get 22188.

$$22188-7x+x=x^{2}$$

Combine -7x and x to get -6x.

$$22188-6x=x^{2}$$

Subtract x^{2} from both sides.

$$22188-6x-x^{2}=0$$

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.

$$-x^{2}-6x+22188=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and 22188 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 22188}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 22188}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 22188}}{2\left(-1\right)}$$

Multiply 4 times 22188.

$$x=\frac{-\left(-6\right)±\sqrt{36+88752}}{2\left(-1\right)}$$

Add 36 to 88752.

$$x=\frac{-\left(-6\right)±\sqrt{88788}}{2\left(-1\right)}$$

Take the square root of 88788.

$$x=\frac{-\left(-6\right)±14\sqrt{453}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±14\sqrt{453}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±14\sqrt{453}}{-2}$$

Now solve the equation x=\frac{6±14\sqrt{453}}{-2} when ± is plus. Add 6 to 14\sqrt{453}.

$$x=\frac{14\sqrt{453}+6}{-2}$$

Divide 6+14\sqrt{453} by -2.

$$x=-7\sqrt{453}-3$$

Now solve the equation x=\frac{6±14\sqrt{453}}{-2} when ± is minus. Subtract 14\sqrt{453} from 6.

$$x=\frac{6-14\sqrt{453}}{-2}$$

Divide 6-14\sqrt{453} by -2.

$$x=7\sqrt{453}-3$$

The equation is now solved.

$$x=-7\sqrt{453}-3 x=7\sqrt{453}-3$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$14-7x+20+25+22129+x=xx$$

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

$$14-7x+20+25+22129+x=x^{2}$$

Add 14 and 20 to get 34.

$$34-7x+25+22129+x=x^{2}$$

Add 34 and 25 to get 59.

$$59-7x+22129+x=x^{2}$$

Add 59 and 22129 to get 22188.

$$22188-7x+x=x^{2}$$

Combine -7x and x to get -6x.

$$22188-6x=x^{2}$$

Subtract x^{2} from both sides.

$$22188-6x-x^{2}=0$$

Subtract 22188 from both sides. Anything subtracted from zero gives its negation.

$$-6x-x^{2}=-22188$$

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.

$$-x^{2}-6x=-22188$$

Divide both sides by -1.

$$\frac{-x^{2}-6x}{-1}=-\frac{22188}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{6}{-1}\right)x=-\frac{22188}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=-\frac{22188}{-1}$$

Divide -22188 by -1.

$$x^{2}+6x=22188$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=22188+3^{2}$$

Square 3.

$$x^{2}+6x+9=22188+9$$

Add 22188 to 9.

$$x^{2}+6x+9=22197$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=22197$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{22197}$$

Simplify.

$$x+3=7\sqrt{453} x+3=-7\sqrt{453}$$

Subtract 3 from both sides of the equation.

$$x=7\sqrt{453}-3 x=-7\sqrt{453}-3$$