Solve 1/x=-x+25

$$\frac{ 1 }{ x } = -x+25$$

Solve for x

$$x = \frac{3 \sqrt{69} + 25}{2} \approx 24.959935794$$

$$x=\frac{25-3\sqrt{69}}{2}\approx 0.040064206$$

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.

$$1=-xx+x\times 25$$

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

$$1=-x^{2}+x\times 25$$

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

$$-x^{2}+x\times 25=1$$

Subtract 1 from both sides.

$$-x^{2}+x\times 25-1=0$$

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.

$$-x^{2}+25x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 25 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-25±\sqrt{625+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-25±\sqrt{625-4}}{2\left(-1\right)}$$

Add 625 to -4.

$$x=\frac{-25±\sqrt{621}}{2\left(-1\right)}$$

Take the square root of 621.

$$x=\frac{-25±3\sqrt{69}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-25±3\sqrt{69}}{-2}$$

Now solve the equation x=\frac{-25±3\sqrt{69}}{-2} when ± is plus. Add -25 to 3\sqrt{69}.

$$x=\frac{3\sqrt{69}-25}{-2}$$

Divide -25+3\sqrt{69} by -2.

$$x=\frac{25-3\sqrt{69}}{2}$$

Now solve the equation x=\frac{-25±3\sqrt{69}}{-2} when ± is minus. Subtract 3\sqrt{69} from -25.

$$x=\frac{-3\sqrt{69}-25}{-2}$$

Divide -25-3\sqrt{69} by -2.

$$x=\frac{3\sqrt{69}+25}{2}$$

The equation is now solved.

$$x=\frac{25-3\sqrt{69}}{2} x=\frac{3\sqrt{69}+25}{2}$$

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.

$$1=-xx+x\times 25$$

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

$$1=-x^{2}+x\times 25$$

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

$$-x^{2}+x\times 25=1$$

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.

$$-x^{2}+25x=1$$

Divide both sides by -1.

$$\frac{-x^{2}+25x}{-1}=\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{25}{-1}x=\frac{1}{-1}$$

Divide 25 by -1.

$$x^{2}-25x=\frac{1}{-1}$$

Divide 1 by -1.

$$x^{2}-25x=-1$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-1+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=-1+\frac{625}{4}$$

Add -1 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{621}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{621}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{621}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{3\sqrt{69}}{2} x-\frac{25}{2}=-\frac{3\sqrt{69}}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=\frac{3\sqrt{69}+25}{2} x=\frac{25-3\sqrt{69}}{2}$$