Solve 1/x=-x+2.5

$$\frac{ 1 }{ x } = -x+2.5$$

Solve for x

$$x=0.5$$

$$x=2$$

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.

$$1=-xx+x\times 2.5$$

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

$$1=-x^{2}+x\times 2.5$$

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

$$-x^{2}+x\times 2.5=1$$

Subtract 1 from both sides.

$$-x^{2}+x\times 2.5-1=0$$

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.

$$-x^{2}+2.5x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2.5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2.5±\sqrt{2.5^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square 2.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-2.5±\sqrt{6.25-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2.5±\sqrt{6.25+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-2.5±\sqrt{6.25-4}}{2\left(-1\right)}$$

Add 6.25 to -4.

$$x=\frac{-2.5±\sqrt{2.25}}{2\left(-1\right)}$$

Take the square root of 2.25.

$$x=\frac{-2.5±\frac{3}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2.5±\frac{3}{2}}{-2}$$

Now solve the equation x=\frac{-2.5±\frac{3}{2}}{-2} when ± is plus. Add -2.5 to \frac{3}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -1 by -2.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{-2.5±\frac{3}{2}}{-2} when ± is minus. Subtract \frac{3}{2} from -2.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=\frac{1}{2} x=2$$

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.

$$1=-xx+x\times 2.5$$

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

$$1=-x^{2}+x\times 2.5$$

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

$$-x^{2}+x\times 2.5=1$$

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.

$$-x^{2}+2.5x=1$$

Divide both sides by -1.

$$\frac{-x^{2}+2.5x}{-1}=\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2.5}{-1}x=\frac{1}{-1}$$

Divide 2.5 by -1.

$$x^{2}-2.5x=\frac{1}{-1}$$

Divide 1 by -1.

$$x^{2}-2.5x=-1$$

Divide -2.5, the coefficient of the x term, by 2 to get -1.25. Then add the square of -1.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2.5x+\left(-1.25\right)^{2}=-1+\left(-1.25\right)^{2}$$

Square -1.25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-2.5x+1.5625=-1+1.5625$$

Add -1 to 1.5625.

$$x^{2}-2.5x+1.5625=0.5625$$

Factor x^{2}-2.5x+1.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1.25\right)^{2}=0.5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1.25\right)^{2}}=\sqrt{0.5625}$$

Simplify.

$$x-1.25=\frac{3}{4} x-1.25=-\frac{3}{4}$$

Add 1.25 to both sides of the equation.

$$x=2 x=\frac{1}{2}$$