Solve x+1/x+0.25=x

$$\frac { x + 1 } { x } + 0.25 = x$$

Solve for x

$$x = \frac{\sqrt{89} + 5}{8} \approx 1.804247642$$

$$x=\frac{5-\sqrt{89}}{8}\approx -0.554247642$$

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.

$$x+1+x\times 0.25=xx$$

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

$$x+1+x\times 0.25=x^{2}$$

Combine x and x\times 0.25 to get 1.25x.

$$1.25x+1=x^{2}$$

Subtract x^{2} from both sides.

$$1.25x+1-x^{2}=0$$

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.

$$-x^{2}+1.25x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1.25 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.25±\sqrt{1.25^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

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

$$x=\frac{-1.25±\sqrt{1.5625-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1.25±\sqrt{1.5625+4}}{2\left(-1\right)}$$

Add 1.5625 to 4.

$$x=\frac{-1.25±\sqrt{5.5625}}{2\left(-1\right)}$$

Take the square root of 5.5625.

$$x=\frac{-1.25±\frac{\sqrt{89}}{4}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1.25±\frac{\sqrt{89}}{4}}{-2}$$

Now solve the equation x=\frac{-1.25±\frac{\sqrt{89}}{4}}{-2} when ± is plus. Add -1.25 to \frac{\sqrt{89}}{4}.

$$x=\frac{\sqrt{89}-5}{-2\times 4}$$

Divide \frac{-5+\sqrt{89}}{4} by -2.

$$x=\frac{5-\sqrt{89}}{8}$$

Now solve the equation x=\frac{-1.25±\frac{\sqrt{89}}{4}}{-2} when ± is minus. Subtract \frac{\sqrt{89}}{4} from -1.25.

$$x=\frac{-\sqrt{89}-5}{-2\times 4}$$

Divide \frac{-5-\sqrt{89}}{4} by -2.

$$x=\frac{\sqrt{89}+5}{8}$$

The equation is now solved.

$$x=\frac{5-\sqrt{89}}{8} x=\frac{\sqrt{89}+5}{8}$$

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.

$$x+1+x\times 0.25=xx$$

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

$$x+1+x\times 0.25=x^{2}$$

Combine x and x\times 0.25 to get 1.25x.

$$1.25x+1=x^{2}$$

Subtract x^{2} from both sides.

$$1.25x+1-x^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$1.25x-x^{2}=-1$$

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.

$$-x^{2}+1.25x=-1$$

Divide both sides by -1.

$$\frac{-x^{2}+1.25x}{-1}=-\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1.25}{-1}x=-\frac{1}{-1}$$

Divide 1.25 by -1.

$$x^{2}-1.25x=-\frac{1}{-1}$$

Divide -1 by -1.

$$x^{2}-1.25x=1$$

Divide -1.25, the coefficient of the x term, by 2 to get -0.625. Then add the square of -0.625 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.25x+\left(-0.625\right)^{2}=1+\left(-0.625\right)^{2}$$

Square -0.625 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.25x+0.390625=1+0.390625$$

Add 1 to 0.390625.

$$x^{2}-1.25x+0.390625=1.390625$$

Factor x^{2}-1.25x+0.390625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.625\right)^{2}=1.390625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.625\right)^{2}}=\sqrt{1.390625}$$

Simplify.

$$x-0.625=\frac{\sqrt{89}}{8} x-0.625=-\frac{\sqrt{89}}{8}$$

Add 0.625 to both sides of the equation.

$$x=\frac{\sqrt{89}+5}{8} x=\frac{5-\sqrt{89}}{8}$$