Solve [x-2]+0.1(x-1)x+0.5x=11

$$[ x - 2 ] + 0.1 ( x - 1 ) x + 0.5 x = 11$$

Solve for x (complex solution)

$$x=\sqrt{179}-7\approx 6.37908816$$

$$x=-\left(\sqrt{179}+7\right)\approx -20.37908816$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.1 by x-1.

$$x-2+\left(0.1x-0.1\right)x+0.5x=11$$

Use the distributive property to multiply 0.1x-0.1 by x.

$$x-2+0.1x^{2}-0.1x+0.5x=11$$

Combine x and -0.1x to get 0.9x.

$$0.9x-2+0.1x^{2}+0.5x=11$$

Combine 0.9x and 0.5x to get 1.4x.

$$1.4x-2+0.1x^{2}=11$$

Subtract 11 from both sides.

$$1.4x-2+0.1x^{2}-11=0$$

Subtract 11 from -2 to get -13.

$$1.4x-13+0.1x^{2}=0$$

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.

$$0.1x^{2}+1.4x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.1 for a, 1.4 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.4±\sqrt{1.4^{2}-4\times 0.1\left(-13\right)}}{2\times 0.1}$$

Square 1.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.4±\sqrt{1.96-4\times 0.1\left(-13\right)}}{2\times 0.1}$$

Multiply -4 times 0.1.

$$x=\frac{-1.4±\sqrt{1.96-0.4\left(-13\right)}}{2\times 0.1}$$

Multiply -0.4 times -13.

$$x=\frac{-1.4±\sqrt{1.96+5.2}}{2\times 0.1}$$

Add 1.96 to 5.2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.4±\sqrt{7.16}}{2\times 0.1}$$

Take the square root of 7.16.

$$x=\frac{-1.4±\frac{\sqrt{179}}{5}}{2\times 0.1}$$

Multiply 2 times 0.1.

$$x=\frac{-1.4±\frac{\sqrt{179}}{5}}{0.2}$$

Now solve the equation x=\frac{-1.4±\frac{\sqrt{179}}{5}}{0.2} when ± is plus. Add -1.4 to \frac{\sqrt{179}}{5}.

$$x=\frac{\sqrt{179}-7}{0.2\times 5}$$

Divide \frac{-7+\sqrt{179}}{5} by 0.2 by multiplying \frac{-7+\sqrt{179}}{5} by the reciprocal of 0.2.

$$x=\sqrt{179}-7$$

Now solve the equation x=\frac{-1.4±\frac{\sqrt{179}}{5}}{0.2} when ± is minus. Subtract \frac{\sqrt{179}}{5} from -1.4.

$$x=\frac{-\sqrt{179}-7}{0.2\times 5}$$

Divide \frac{-7-\sqrt{179}}{5} by 0.2 by multiplying \frac{-7-\sqrt{179}}{5} by the reciprocal of 0.2.

$$x=-\sqrt{179}-7$$

The equation is now solved.

$$x=\sqrt{179}-7 x=-\sqrt{179}-7$$

Steps for Completing the Square

Use the distributive property to multiply 0.1 by x-1.

$$x-2+\left(0.1x-0.1\right)x+0.5x=11$$

Use the distributive property to multiply 0.1x-0.1 by x.

$$x-2+0.1x^{2}-0.1x+0.5x=11$$

Combine x and -0.1x to get 0.9x.

$$0.9x-2+0.1x^{2}+0.5x=11$$

Combine 0.9x and 0.5x to get 1.4x.

$$1.4x-2+0.1x^{2}=11$$

Add 2 to both sides.

$$1.4x+0.1x^{2}=11+2$$

Add 11 and 2 to get 13.

$$1.4x+0.1x^{2}=13$$

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.

$$0.1x^{2}+1.4x=13$$

Multiply both sides by 10.

$$\frac{0.1x^{2}+1.4x}{0.1}=\frac{13}{0.1}$$

Dividing by 0.1 undoes the multiplication by 0.1.

$$x^{2}+\frac{1.4}{0.1}x=\frac{13}{0.1}$$

Divide 1.4 by 0.1 by multiplying 1.4 by the reciprocal of 0.1.

$$x^{2}+14x=\frac{13}{0.1}$$

Divide 13 by 0.1 by multiplying 13 by the reciprocal of 0.1.

$$x^{2}+14x=130$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=130+7^{2}$$

Square 7.

$$x^{2}+14x+49=130+49$$

Add 130 to 49.

$$x^{2}+14x+49=179$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=179$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{179}$$

Simplify.

$$x+7=\sqrt{179} x+7=-\sqrt{179}$$

Subtract 7 from both sides of the equation.

$$x=\sqrt{179}-7 x=-\sqrt{179}-7$$

Solve for x

$$x=\sqrt{179}-7\approx 6.37908816$$

$$x=-\sqrt{179}-7\approx -20.37908816$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.1 by x-1.

$$x-2+\left(0.1x-0.1\right)x+0.5x=11$$

Use the distributive property to multiply 0.1x-0.1 by x.

$$x-2+0.1x^{2}-0.1x+0.5x=11$$

Combine x and -0.1x to get 0.9x.

$$0.9x-2+0.1x^{2}+0.5x=11$$

Combine 0.9x and 0.5x to get 1.4x.

$$1.4x-2+0.1x^{2}=11$$

Subtract 11 from both sides.

$$1.4x-2+0.1x^{2}-11=0$$

Subtract 11 from -2 to get -13.

$$1.4x-13+0.1x^{2}=0$$

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.

$$0.1x^{2}+1.4x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.1 for a, 1.4 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.4±\sqrt{1.4^{2}-4\times 0.1\left(-13\right)}}{2\times 0.1}$$

Square 1.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.4±\sqrt{1.96-4\times 0.1\left(-13\right)}}{2\times 0.1}$$

Multiply -4 times 0.1.

$$x=\frac{-1.4±\sqrt{1.96-0.4\left(-13\right)}}{2\times 0.1}$$

Multiply -0.4 times -13.

$$x=\frac{-1.4±\sqrt{1.96+5.2}}{2\times 0.1}$$

Add 1.96 to 5.2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.4±\sqrt{7.16}}{2\times 0.1}$$

Take the square root of 7.16.

$$x=\frac{-1.4±\frac{\sqrt{179}}{5}}{2\times 0.1}$$

Multiply 2 times 0.1.

$$x=\frac{-1.4±\frac{\sqrt{179}}{5}}{0.2}$$

Now solve the equation x=\frac{-1.4±\frac{\sqrt{179}}{5}}{0.2} when ± is plus. Add -1.4 to \frac{\sqrt{179}}{5}.

$$x=\frac{\sqrt{179}-7}{0.2\times 5}$$

Divide \frac{-7+\sqrt{179}}{5} by 0.2 by multiplying \frac{-7+\sqrt{179}}{5} by the reciprocal of 0.2.

$$x=\sqrt{179}-7$$

Now solve the equation x=\frac{-1.4±\frac{\sqrt{179}}{5}}{0.2} when ± is minus. Subtract \frac{\sqrt{179}}{5} from -1.4.

$$x=\frac{-\sqrt{179}-7}{0.2\times 5}$$

Divide \frac{-7-\sqrt{179}}{5} by 0.2 by multiplying \frac{-7-\sqrt{179}}{5} by the reciprocal of 0.2.

$$x=-\sqrt{179}-7$$

The equation is now solved.

$$x=\sqrt{179}-7 x=-\sqrt{179}-7$$

Steps for Completing the Square

Use the distributive property to multiply 0.1 by x-1.

$$x-2+\left(0.1x-0.1\right)x+0.5x=11$$

Use the distributive property to multiply 0.1x-0.1 by x.

$$x-2+0.1x^{2}-0.1x+0.5x=11$$

Combine x and -0.1x to get 0.9x.

$$0.9x-2+0.1x^{2}+0.5x=11$$

Combine 0.9x and 0.5x to get 1.4x.

$$1.4x-2+0.1x^{2}=11$$

Add 2 to both sides.

$$1.4x+0.1x^{2}=11+2$$

Add 11 and 2 to get 13.

$$1.4x+0.1x^{2}=13$$

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.

$$0.1x^{2}+1.4x=13$$

Multiply both sides by 10.

$$\frac{0.1x^{2}+1.4x}{0.1}=\frac{13}{0.1}$$

Dividing by 0.1 undoes the multiplication by 0.1.

$$x^{2}+\frac{1.4}{0.1}x=\frac{13}{0.1}$$

Divide 1.4 by 0.1 by multiplying 1.4 by the reciprocal of 0.1.

$$x^{2}+14x=\frac{13}{0.1}$$

Divide 13 by 0.1 by multiplying 13 by the reciprocal of 0.1.

$$x^{2}+14x=130$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=130+7^{2}$$

Square 7.

$$x^{2}+14x+49=130+49$$

Add 130 to 49.

$$x^{2}+14x+49=179$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=179$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{179}$$

Simplify.

$$x+7=\sqrt{179} x+7=-\sqrt{179}$$

Subtract 7 from both sides of the equation.

$$x=\sqrt{179}-7 x=-\sqrt{179}-7$$