Solve 144x^2-128x+64=256

$$144 { x }^{ 2 } -128x+64=256$$

Solve for x

$$x = \frac{2 \sqrt{31} + 4}{9} \approx 1.681725414$$

$$x=\frac{4-2\sqrt{31}}{9}\approx -0.792836525$$

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.

$$144x^{2}-128x+64=256$$

Subtract 256 from both sides of the equation.

$$144x^{2}-128x+64-256=256-256$$

Subtracting 256 from itself leaves 0.

$$144x^{2}-128x+64-256=0$$

Subtract 256 from 64.

$$144x^{2}-128x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 144 for a, -128 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-128\right)±\sqrt{\left(-128\right)^{2}-4\times 144\left(-192\right)}}{2\times 144}$$

Square -128.

$$x=\frac{-\left(-128\right)±\sqrt{16384-4\times 144\left(-192\right)}}{2\times 144}$$

Multiply -4 times 144.

$$x=\frac{-\left(-128\right)±\sqrt{16384-576\left(-192\right)}}{2\times 144}$$

Multiply -576 times -192.

$$x=\frac{-\left(-128\right)±\sqrt{16384+110592}}{2\times 144}$$

Add 16384 to 110592.

$$x=\frac{-\left(-128\right)±\sqrt{126976}}{2\times 144}$$

Take the square root of 126976.

$$x=\frac{-\left(-128\right)±64\sqrt{31}}{2\times 144}$$

The opposite of -128 is 128.

$$x=\frac{128±64\sqrt{31}}{2\times 144}$$

Multiply 2 times 144.

$$x=\frac{128±64\sqrt{31}}{288}$$

Now solve the equation x=\frac{128±64\sqrt{31}}{288} when ± is plus. Add 128 to 64\sqrt{31}.

$$x=\frac{64\sqrt{31}+128}{288}$$

Divide 128+64\sqrt{31} by 288.

$$x=\frac{2\sqrt{31}+4}{9}$$

Now solve the equation x=\frac{128±64\sqrt{31}}{288} when ± is minus. Subtract 64\sqrt{31} from 128.

$$x=\frac{128-64\sqrt{31}}{288}$$

Divide 128-64\sqrt{31} by 288.

$$x=\frac{4-2\sqrt{31}}{9}$$

The equation is now solved.

$$x=\frac{2\sqrt{31}+4}{9} x=\frac{4-2\sqrt{31}}{9}$$

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.

$$144x^{2}-128x+64=256$$

Subtract 64 from both sides of the equation.

$$144x^{2}-128x+64-64=256-64$$

Subtracting 64 from itself leaves 0.

$$144x^{2}-128x=256-64$$

Subtract 64 from 256.

$$144x^{2}-128x=192$$

Divide both sides by 144.

$$\frac{144x^{2}-128x}{144}=\frac{192}{144}$$

Dividing by 144 undoes the multiplication by 144.

$$x^{2}+\left(-\frac{128}{144}\right)x=\frac{192}{144}$$

Reduce the fraction \frac{-128}{144} to lowest terms by extracting and canceling out 16.

$$x^{2}-\frac{8}{9}x=\frac{192}{144}$$

Reduce the fraction \frac{192}{144} to lowest terms by extracting and canceling out 48.

$$x^{2}-\frac{8}{9}x=\frac{4}{3}$$

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

$$x^{2}-\frac{8}{9}x+\left(-\frac{4}{9}\right)^{2}=\frac{4}{3}+\left(-\frac{4}{9}\right)^{2}$$

Square -\frac{4}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{4}{3}+\frac{16}{81}$$

Add \frac{4}{3} to \frac{16}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{124}{81}$$

Factor x^{2}-\frac{8}{9}x+\frac{16}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{9}\right)^{2}=\frac{124}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{9}\right)^{2}}=\sqrt{\frac{124}{81}}$$

Simplify.

$$x-\frac{4}{9}=\frac{2\sqrt{31}}{9} x-\frac{4}{9}=-\frac{2\sqrt{31}}{9}$$

Add \frac{4}{9} to both sides of the equation.

$$x=\frac{2\sqrt{31}+4}{9} x=\frac{4-2\sqrt{31}}{9}$$