Solve x^2=9(0.02-x)^2

$$x ^ { 2 } = 9 ( 0.02 - x ) ^ { 2 }$$

Solve for x

$$x=0.03$$

$$x=0.015$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.02-x\right)^{2}.

$$x^{2}=9\left(0.0004-0.04x+x^{2}\right)$$

Use the distributive property to multiply 9 by 0.0004-0.04x+x^{2}.

$$x^{2}=0.0036-0.36x+9x^{2}$$

Subtract 0.0036 from both sides.

$$x^{2}-0.0036=-0.36x+9x^{2}$$

Add 0.36x to both sides.

$$x^{2}-0.0036+0.36x=9x^{2}$$

Subtract 9x^{2} from both sides.

$$x^{2}-0.0036+0.36x-9x^{2}=0$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$-8x^{2}-0.0036+0.36x=0$$

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.

$$-8x^{2}+0.36x-0.0036=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 0.36 for b, and -0.0036 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.36±\sqrt{0.36^{2}-4\left(-8\right)\left(-0.0036\right)}}{2\left(-8\right)}$$

Square 0.36 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.36±\sqrt{0.1296-4\left(-8\right)\left(-0.0036\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-0.36±\sqrt{0.1296+32\left(-0.0036\right)}}{2\left(-8\right)}$$

Multiply 32 times -0.0036.

$$x=\frac{-0.36±\sqrt{\frac{81-72}{625}}}{2\left(-8\right)}$$

Add 0.1296 to -0.1152 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.36±\sqrt{0.0144}}{2\left(-8\right)}$$

Take the square root of 0.0144.

$$x=\frac{-0.36±\frac{3}{25}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-0.36±\frac{3}{25}}{-16}$$

Now solve the equation x=\frac{-0.36±\frac{3}{25}}{-16} when ± is plus. Add -0.36 to \frac{3}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{6}{25}}{-16}$$

Divide -\frac{6}{25} by -16.

$$x=\frac{3}{200}$$

Now solve the equation x=\frac{-0.36±\frac{3}{25}}{-16} when ± is minus. Subtract \frac{3}{25} from -0.36 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{12}{25}}{-16}$$

Divide -\frac{12}{25} by -16.

$$x=\frac{3}{100}$$

The equation is now solved.

$$x=\frac{3}{200} x=\frac{3}{100}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.02-x\right)^{2}.

$$x^{2}=9\left(0.0004-0.04x+x^{2}\right)$$

Use the distributive property to multiply 9 by 0.0004-0.04x+x^{2}.

$$x^{2}=0.0036-0.36x+9x^{2}$$

Add 0.36x to both sides.

$$x^{2}+0.36x=0.0036+9x^{2}$$

Subtract 9x^{2} from both sides.

$$x^{2}+0.36x-9x^{2}=0.0036$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$-8x^{2}+0.36x=0.0036$$

Divide both sides by -8.

$$\frac{-8x^{2}+0.36x}{-8}=\frac{0.0036}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{0.36}{-8}x=\frac{0.0036}{-8}$$

Divide 0.36 by -8.

$$x^{2}-0.045x=\frac{0.0036}{-8}$$

Divide 0.0036 by -8.

$$x^{2}-0.045x=-0.00045$$

Divide -0.045, the coefficient of the x term, by 2 to get -0.0225. Then add the square of -0.0225 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.045x+\left(-0.0225\right)^{2}=-0.00045+\left(-0.0225\right)^{2}$$

Square -0.0225 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.045x+0.00050625=-0.00045+0.00050625$$

Add -0.00045 to 0.00050625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.045x+0.00050625=0.00005625$$

Factor x^{2}-0.045x+0.00050625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.0225\right)^{2}=0.00005625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.0225\right)^{2}}=\sqrt{0.00005625}$$

Simplify.

$$x-0.0225=\frac{3}{400} x-0.0225=-\frac{3}{400}$$

Add 0.0225 to both sides of the equation.

$$x=\frac{3}{100} x=\frac{3}{200}$$