Solve 900x^2-136x+4=0

$$900 { x }^{ 2 } -136x+4=0$$

Solve for x

$$x=\frac{1}{25}=0.04$$

$$x=\frac{1}{9}\approx 0.111111111$$

Steps Using the Quadratic Formula

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.

$$900x^{2}-136x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 900 for a, -136 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-136\right)±\sqrt{\left(-136\right)^{2}-4\times 900\times 4}}{2\times 900}$$

Square -136.

$$x=\frac{-\left(-136\right)±\sqrt{18496-4\times 900\times 4}}{2\times 900}$$

Multiply -4 times 900.

$$x=\frac{-\left(-136\right)±\sqrt{18496-3600\times 4}}{2\times 900}$$

Multiply -3600 times 4.

$$x=\frac{-\left(-136\right)±\sqrt{18496-14400}}{2\times 900}$$

Add 18496 to -14400.

$$x=\frac{-\left(-136\right)±\sqrt{4096}}{2\times 900}$$

Take the square root of 4096.

$$x=\frac{-\left(-136\right)±64}{2\times 900}$$

The opposite of -136 is 136.

$$x=\frac{136±64}{2\times 900}$$

Multiply 2 times 900.

$$x=\frac{136±64}{1800}$$

Now solve the equation x=\frac{136±64}{1800} when ± is plus. Add 136 to 64.

$$x=\frac{200}{1800}$$

Reduce the fraction \frac{200}{1800} to lowest terms by extracting and canceling out 200.

$$x=\frac{1}{9}$$

Now solve the equation x=\frac{136±64}{1800} when ± is minus. Subtract 64 from 136.

$$x=\frac{72}{1800}$$

Reduce the fraction \frac{72}{1800} to lowest terms by extracting and canceling out 72.

$$x=\frac{1}{25}$$

The equation is now solved.

$$x=\frac{1}{9} x=\frac{1}{25}$$

Steps for Completing the Square

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.

$$900x^{2}-136x+4=0$$

Subtract 4 from both sides of the equation.

$$900x^{2}-136x+4-4=-4$$

Subtracting 4 from itself leaves 0.

$$900x^{2}-136x=-4$$

Divide both sides by 900.

$$\frac{900x^{2}-136x}{900}=-\frac{4}{900}$$

Dividing by 900 undoes the multiplication by 900.

$$x^{2}+\left(-\frac{136}{900}\right)x=-\frac{4}{900}$$

Reduce the fraction \frac{-136}{900} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{34}{225}x=-\frac{4}{900}$$

Reduce the fraction \frac{-4}{900} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{34}{225}x=-\frac{1}{225}$$

Divide -\frac{34}{225}, the coefficient of the x term, by 2 to get -\frac{17}{225}. Then add the square of -\frac{17}{225} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{34}{225}x+\left(-\frac{17}{225}\right)^{2}=-\frac{1}{225}+\left(-\frac{17}{225}\right)^{2}$$

Square -\frac{17}{225} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{34}{225}x+\frac{289}{50625}=-\frac{1}{225}+\frac{289}{50625}$$

Add -\frac{1}{225} to \frac{289}{50625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{34}{225}x+\frac{289}{50625}=\frac{64}{50625}$$

Factor x^{2}-\frac{34}{225}x+\frac{289}{50625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{225}\right)^{2}=\frac{64}{50625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{225}\right)^{2}}=\sqrt{\frac{64}{50625}}$$

Simplify.

$$x-\frac{17}{225}=\frac{8}{225} x-\frac{17}{225}=-\frac{8}{225}$$

Add \frac{17}{225} to both sides of the equation.

$$x=\frac{1}{9} x=\frac{1}{25}$$