Solve 0.10x+0.05x(x-5)=2.15

$$0.10x+0.05x(x-5)=2.15$$

Solve for x

$$x = \frac{\sqrt{181} + 3}{2} \approx 8.226812024$$

$$x=\frac{3-\sqrt{181}}{2}\approx -5.226812024$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.05x by x-5.

$$0.1x+0.05x^{2}-0.25x=2.15$$

Combine 0.1x and -0.25x to get -0.15x.

$$-0.15x+0.05x^{2}=2.15$$

Subtract 2.15 from both sides.

$$-0.15x+0.05x^{2}-2.15=0$$

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.

$$0.05x^{2}-0.15x-2.15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.05 for a, -0.15 for b, and -2.15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.15\right)±\sqrt{\left(-0.15\right)^{2}-4\times 0.05\left(-2.15\right)}}{2\times 0.05}$$

Square -0.15 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.15\right)±\sqrt{0.0225-4\times 0.05\left(-2.15\right)}}{2\times 0.05}$$

Multiply -4 times 0.05.

$$x=\frac{-\left(-0.15\right)±\sqrt{0.0225-0.2\left(-2.15\right)}}{2\times 0.05}$$

Multiply -0.2 times -2.15 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.15\right)±\sqrt{0.0225+0.43}}{2\times 0.05}$$

Add 0.0225 to 0.43 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.15\right)±\sqrt{0.4525}}{2\times 0.05}$$

Take the square root of 0.4525.

$$x=\frac{-\left(-0.15\right)±\frac{\sqrt{181}}{20}}{2\times 0.05}$$

The opposite of -0.15 is 0.15.

$$x=\frac{0.15±\frac{\sqrt{181}}{20}}{2\times 0.05}$$

Multiply 2 times 0.05.

$$x=\frac{0.15±\frac{\sqrt{181}}{20}}{0.1}$$

Now solve the equation x=\frac{0.15±\frac{\sqrt{181}}{20}}{0.1} when ± is plus. Add 0.15 to \frac{\sqrt{181}}{20}.

$$x=\frac{\sqrt{181}+3}{0.1\times 20}$$

Divide \frac{3+\sqrt{181}}{20} by 0.1 by multiplying \frac{3+\sqrt{181}}{20} by the reciprocal of 0.1.

$$x=\frac{\sqrt{181}+3}{2}$$

Now solve the equation x=\frac{0.15±\frac{\sqrt{181}}{20}}{0.1} when ± is minus. Subtract \frac{\sqrt{181}}{20} from 0.15.

$$x=\frac{3-\sqrt{181}}{0.1\times 20}$$

Divide \frac{3-\sqrt{181}}{20} by 0.1 by multiplying \frac{3-\sqrt{181}}{20} by the reciprocal of 0.1.

$$x=\frac{3-\sqrt{181}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{181}+3}{2} x=\frac{3-\sqrt{181}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 0.05x by x-5.

$$0.1x+0.05x^{2}-0.25x=2.15$$

Combine 0.1x and -0.25x to get -0.15x.

$$-0.15x+0.05x^{2}=2.15$$

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.

$$0.05x^{2}-0.15x=2.15$$

Multiply both sides by 20.

$$\frac{0.05x^{2}-0.15x}{0.05}=\frac{2.15}{0.05}$$

Dividing by 0.05 undoes the multiplication by 0.05.

$$x^{2}+\left(-\frac{0.15}{0.05}\right)x=\frac{2.15}{0.05}$$

Divide -0.15 by 0.05 by multiplying -0.15 by the reciprocal of 0.05.

$$x^{2}-3x=\frac{2.15}{0.05}$$

Divide 2.15 by 0.05 by multiplying 2.15 by the reciprocal of 0.05.

$$x^{2}-3x=43$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=43+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=43+\frac{9}{4}$$

Add 43 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{181}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{181}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{181}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{181}}{2} x-\frac{3}{2}=-\frac{\sqrt{181}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{181}+3}{2} x=\frac{3-\sqrt{181}}{2}$$