Solve xdiv10+2xdiv10x=0.06x+3

$$x \div 10+2x \div 10x=0.06x+3$$

Solve for x

$$x = \frac{\sqrt{1501} - 1}{10} \approx 3.774274126$$

$$x=\frac{-\sqrt{1501}-1}{10}\approx -3.974274126$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10.

$$x+2xx=0.6x+30$$

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

$$x+2x^{2}=0.6x+30$$

Subtract 0.6x from both sides.

$$x+2x^{2}-0.6x=30$$

Combine x and -0.6x to get 0.4x.

$$0.4x+2x^{2}=30$$

Subtract 30 from both sides.

$$0.4x+2x^{2}-30=0$$

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.

$$2x^{2}+0.4x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0.4 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.4±\sqrt{0.4^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Square 0.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.4±\sqrt{0.16-4\times 2\left(-30\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-0.4±\sqrt{0.16-8\left(-30\right)}}{2\times 2}$$

Multiply -8 times -30.

$$x=\frac{-0.4±\sqrt{0.16+240}}{2\times 2}$$

Add 0.16 to 240.

$$x=\frac{-0.4±\sqrt{240.16}}{2\times 2}$$

Take the square root of 240.16.

$$x=\frac{-0.4±\frac{2\sqrt{1501}}{5}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-0.4±\frac{2\sqrt{1501}}{5}}{4}$$

Now solve the equation x=\frac{-0.4±\frac{2\sqrt{1501}}{5}}{4} when ± is plus. Add -0.4 to \frac{2\sqrt{1501}}{5}.

$$x=\frac{2\sqrt{1501}-2}{4\times 5}$$

Divide \frac{-2+2\sqrt{1501}}{5} by 4.

$$x=\frac{\sqrt{1501}-1}{10}$$

Now solve the equation x=\frac{-0.4±\frac{2\sqrt{1501}}{5}}{4} when ± is minus. Subtract \frac{2\sqrt{1501}}{5} from -0.4.

$$x=\frac{-2\sqrt{1501}-2}{4\times 5}$$

Divide \frac{-2-2\sqrt{1501}}{5} by 4.

$$x=\frac{-\sqrt{1501}-1}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{1501}-1}{10} x=\frac{-\sqrt{1501}-1}{10}$$

Steps for Completing the Square

Multiply both sides of the equation by 10.

$$x+2xx=0.6x+30$$

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

$$x+2x^{2}=0.6x+30$$

Subtract 0.6x from both sides.

$$x+2x^{2}-0.6x=30$$

Combine x and -0.6x to get 0.4x.

$$0.4x+2x^{2}=30$$

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.

$$2x^{2}+0.4x=30$$

Divide both sides by 2.

$$\frac{2x^{2}+0.4x}{2}=\frac{30}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{0.4}{2}x=\frac{30}{2}$$

Divide 0.4 by 2.

$$x^{2}+0.2x=\frac{30}{2}$$

Divide 30 by 2.

$$x^{2}+0.2x=15$$

Divide 0.2, the coefficient of the x term, by 2 to get 0.1. Then add the square of 0.1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.2x+0.1^{2}=15+0.1^{2}$$

Square 0.1 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.2x+0.01=15+0.01$$

Add 15 to 0.01.

$$x^{2}+0.2x+0.01=15.01$$

Factor x^{2}+0.2x+0.01. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.1\right)^{2}=15.01$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.1\right)^{2}}=\sqrt{15.01}$$

Simplify.

$$x+0.1=\frac{\sqrt{1501}}{10} x+0.1=-\frac{\sqrt{1501}}{10}$$

Subtract 0.1 from both sides of the equation.

$$x=\frac{\sqrt{1501}-1}{10} x=\frac{-\sqrt{1501}-1}{10}$$