Solve left(1.18-xright)^2=0.8x

$${ \left(1.18-x \right) }^{ 2 } =0.8x$$

Solve for x

$$x=\frac{\sqrt{690}}{25}+1.58\approx 2.630714043$$

$$x=-\frac{\sqrt{690}}{25}+1.58\approx 0.529285957$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1.18-x\right)^{2}.

$$1.3924-2.36x+x^{2}=0.8x$$

Subtract 0.8x from both sides.

$$1.3924-2.36x+x^{2}-0.8x=0$$

Combine -2.36x and -0.8x to get -3.16x.

$$1.3924-3.16x+x^{2}=0$$

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.

$$x^{2}-3.16x+1.3924=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3.16 for b, and 1.3924 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3.16\right)±\sqrt{\left(-3.16\right)^{2}-4\times 1.3924}}{2}$$

Square -3.16 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-3.16\right)±\sqrt{9.9856-4\times 1.3924}}{2}$$

Multiply -4 times 1.3924.

$$x=\frac{-\left(-3.16\right)±\sqrt{\frac{6241-3481}{625}}}{2}$$

Add 9.9856 to -5.5696 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-3.16\right)±\sqrt{4.416}}{2}$$

Take the square root of 4.416.

$$x=\frac{-\left(-3.16\right)±\frac{2\sqrt{690}}{25}}{2}$$

The opposite of -3.16 is 3.16.

$$x=\frac{3.16±\frac{2\sqrt{690}}{25}}{2}$$

Now solve the equation x=\frac{3.16±\frac{2\sqrt{690}}{25}}{2} when ± is plus. Add 3.16 to \frac{2\sqrt{690}}{25}.

$$x=\frac{2\sqrt{690}+79}{2\times 25}$$

Divide \frac{79+2\sqrt{690}}{25} by 2.

$$x=\frac{\sqrt{690}}{25}+\frac{79}{50}$$

Now solve the equation x=\frac{3.16±\frac{2\sqrt{690}}{25}}{2} when ± is minus. Subtract \frac{2\sqrt{690}}{25} from 3.16.

$$x=\frac{79-2\sqrt{690}}{2\times 25}$$

Divide \frac{79-2\sqrt{690}}{25} by 2.

$$x=-\frac{\sqrt{690}}{25}+\frac{79}{50}$$

The equation is now solved.

$$x=\frac{\sqrt{690}}{25}+\frac{79}{50} x=-\frac{\sqrt{690}}{25}+\frac{79}{50}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1.18-x\right)^{2}.

$$1.3924-2.36x+x^{2}=0.8x$$

Subtract 0.8x from both sides.

$$1.3924-2.36x+x^{2}-0.8x=0$$

Combine -2.36x and -0.8x to get -3.16x.

$$1.3924-3.16x+x^{2}=0$$

Subtract 1.3924 from both sides. Anything subtracted from zero gives its negation.

$$-3.16x+x^{2}=-1.3924$$

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.

$$x^{2}-3.16x=-1.3924$$

Divide -3.16, the coefficient of the x term, by 2 to get -1.58. Then add the square of -1.58 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3.16x+\left(-1.58\right)^{2}=-1.3924+\left(-1.58\right)^{2}$$

Square -1.58 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3.16x+2.4964=\frac{-3481+6241}{2500}$$

Add -1.3924 to 2.4964 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3.16x+2.4964=1.104$$

Factor x^{2}-3.16x+2.4964. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1.58\right)^{2}=1.104$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1.58\right)^{2}}=\sqrt{1.104}$$

Simplify.

$$x-1.58=\frac{\sqrt{690}}{25} x-1.58=-\frac{\sqrt{690}}{25}$$

Add 1.58 to both sides of the equation.

$$x=\frac{\sqrt{690}}{25}+\frac{79}{50} x=-\frac{\sqrt{690}}{25}+\frac{79}{50}$$