Solve (1-x-0.523)^2=quad1.1x-x^2

$$( 1 - x - 0.523 ) ^ { 2 } = \quad 1.1 x - x ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{599671}+1027}{2000}\approx 0.900692136$$

$$x=\frac{1027-\sqrt{599671}}{2000}\approx 0.126307864$$

Steps Using the Quadratic Formula

Subtract 0.523 from 1 to get 0.477.

$$\left(0.477-x\right)^{2}=1.1x-x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.477-x\right)^{2}.

$$0.227529-0.954x+x^{2}=1.1x-x^{2}$$

Subtract 1.1x from both sides.

$$0.227529-0.954x+x^{2}-1.1x=-x^{2}$$

Combine -0.954x and -1.1x to get -2.054x.

$$0.227529-2.054x+x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$0.227529-2.054x+x^{2}+x^{2}=0$$

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

$$0.227529-2.054x+2x^{2}=0$$

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.

$$2x^{2}-2.054x+0.227529=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2.054 for b, and 0.227529 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2.054\right)±\sqrt{\left(-2.054\right)^{2}-4\times 2\times 0.227529}}{2\times 2}$$

Square -2.054 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-2.054\right)±\sqrt{4.218916-4\times 2\times 0.227529}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2.054\right)±\sqrt{4.218916-8\times 0.227529}}{2\times 2}$$

Multiply -8 times 0.227529.

$$x=\frac{-\left(-2.054\right)±\sqrt{4.218916-1.820232}}{2\times 2}$$

Add 4.218916 to -1.820232 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-2.054\right)±\sqrt{2.398684}}{2\times 2}$$

Take the square root of 2.398684.

$$x=\frac{-\left(-2.054\right)±\frac{\sqrt{599671}}{500}}{2\times 2}$$

The opposite of -2.054 is 2.054.

$$x=\frac{2.054±\frac{\sqrt{599671}}{500}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2.054±\frac{\sqrt{599671}}{500}}{4}$$

Now solve the equation x=\frac{2.054±\frac{\sqrt{599671}}{500}}{4} when ± is plus. Add 2.054 to \frac{\sqrt{599671}}{500}.

$$x=\frac{\sqrt{599671}+1027}{4\times 500}$$

Divide \frac{1027+\sqrt{599671}}{500} by 4.

$$x=\frac{\sqrt{599671}+1027}{2000}$$

Now solve the equation x=\frac{2.054±\frac{\sqrt{599671}}{500}}{4} when ± is minus. Subtract \frac{\sqrt{599671}}{500} from 2.054.

$$x=\frac{1027-\sqrt{599671}}{4\times 500}$$

Divide \frac{1027-\sqrt{599671}}{500} by 4.

$$x=\frac{1027-\sqrt{599671}}{2000}$$

The equation is now solved.

$$x=\frac{\sqrt{599671}+1027}{2000} x=\frac{1027-\sqrt{599671}}{2000}$$

Steps for Completing the Square

Subtract 0.523 from 1 to get 0.477.

$$\left(0.477-x\right)^{2}=1.1x-x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.477-x\right)^{2}.

$$0.227529-0.954x+x^{2}=1.1x-x^{2}$$

Subtract 1.1x from both sides.

$$0.227529-0.954x+x^{2}-1.1x=-x^{2}$$

Combine -0.954x and -1.1x to get -2.054x.

$$0.227529-2.054x+x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$0.227529-2.054x+x^{2}+x^{2}=0$$

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

$$0.227529-2.054x+2x^{2}=0$$

Subtract 0.227529 from both sides. Anything subtracted from zero gives its negation.

$$-2.054x+2x^{2}=-0.227529$$

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.

$$2x^{2}-2.054x=-0.227529$$

Divide both sides by 2.

$$\frac{2x^{2}-2.054x}{2}=-\frac{0.227529}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{2.054}{2}\right)x=-\frac{0.227529}{2}$$

Divide -2.054 by 2.

$$x^{2}-1.027x=-\frac{0.227529}{2}$$

Divide -0.227529 by 2.

$$x^{2}-1.027x=-0.1137645$$

Divide -1.027, the coefficient of the x term, by 2 to get -0.5135. Then add the square of -0.5135 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.027x+\left(-0.5135\right)^{2}=-0.1137645+\left(-0.5135\right)^{2}$$

Square -0.5135 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.027x+0.26368225=-0.1137645+0.26368225$$

Add -0.1137645 to 0.26368225 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-1.027x+0.26368225=0.14991775$$

Factor x^{2}-1.027x+0.26368225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.5135\right)^{2}=0.14991775$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.5135\right)^{2}}=\sqrt{0.14991775}$$

Simplify.

$$x-0.5135=\frac{\sqrt{599671}}{2000} x-0.5135=-\frac{\sqrt{599671}}{2000}$$

Add 0.5135 to both sides of the equation.

$$x=\frac{\sqrt{599671}+1027}{2000} x=\frac{1027-\sqrt{599671}}{2000}$$