Solve 4(1-a)^2=a

$$4 ( 1 - a ) ^ { 2 } = a$$

Solve for a

$$a = \frac{\sqrt{17} + 9}{8} \approx 1.640388203$$

$$a=\frac{9-\sqrt{17}}{8}\approx 0.609611797$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-a\right)^{2}.

$$4\left(1-2a+a^{2}\right)=a$$

Use the distributive property to multiply 4 by 1-2a+a^{2}.

$$4-8a+4a^{2}=a$$

Subtract a from both sides.

$$4-8a+4a^{2}-a=0$$

Combine -8a and -a to get -9a.

$$4-9a+4a^{2}=0$$

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.

$$4a^{2}-9a+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -9 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\times 4}}{2\times 4}$$

Square -9.

$$a=\frac{-\left(-9\right)±\sqrt{81-4\times 4\times 4}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-9\right)±\sqrt{81-16\times 4}}{2\times 4}$$

Multiply -16 times 4.

$$a=\frac{-\left(-9\right)±\sqrt{81-64}}{2\times 4}$$

Add 81 to -64.

$$a=\frac{-\left(-9\right)±\sqrt{17}}{2\times 4}$$

The opposite of -9 is 9.

$$a=\frac{9±\sqrt{17}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{9±\sqrt{17}}{8}$$

Now solve the equation a=\frac{9±\sqrt{17}}{8} when ± is plus. Add 9 to \sqrt{17}.

$$a=\frac{\sqrt{17}+9}{8}$$

Now solve the equation a=\frac{9±\sqrt{17}}{8} when ± is minus. Subtract \sqrt{17} from 9.

$$a=\frac{9-\sqrt{17}}{8}$$

The equation is now solved.

$$a=\frac{\sqrt{17}+9}{8} a=\frac{9-\sqrt{17}}{8}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-a\right)^{2}.

$$4\left(1-2a+a^{2}\right)=a$$

Use the distributive property to multiply 4 by 1-2a+a^{2}.

$$4-8a+4a^{2}=a$$

Subtract a from both sides.

$$4-8a+4a^{2}-a=0$$

Combine -8a and -a to get -9a.

$$4-9a+4a^{2}=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-9a+4a^{2}=-4$$

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.

$$4a^{2}-9a=-4$$

Divide both sides by 4.

$$\frac{4a^{2}-9a}{4}=-\frac{4}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$a^{2}-\frac{9}{4}a=-\frac{4}{4}$$

Divide -4 by 4.

$$a^{2}-\frac{9}{4}a=-1$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{9}{4}a+\left(-\frac{9}{8}\right)^{2}=-1+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{9}{4}a+\frac{81}{64}=-1+\frac{81}{64}$$

Add -1 to \frac{81}{64}.

$$a^{2}-\frac{9}{4}a+\frac{81}{64}=\frac{17}{64}$$

Factor a^{2}-\frac{9}{4}a+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{9}{8}\right)^{2}=\frac{17}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{9}{8}\right)^{2}}=\sqrt{\frac{17}{64}}$$

Simplify.

$$a-\frac{9}{8}=\frac{\sqrt{17}}{8} a-\frac{9}{8}=-\frac{\sqrt{17}}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$a=\frac{\sqrt{17}+9}{8} a=\frac{9-\sqrt{17}}{8}$$