Solve left(16xright)^2-72x+32=0

$${ \left(16x \right) }^{ 2 } -72x+32 = 0$$

Solve for x (complex solution)

$$x=\frac{9+\sqrt{431}i}{64}\approx 0.140625+0.32438343i$$

$$x=\frac{-\sqrt{431}i+9}{64}\approx 0.140625-0.32438343i$$

Steps Using the Quadratic Formula

Expand \left(16x\right)^{2}.

$$16^{2}x^{2}-72x+32=0$$

Calculate 16 to the power of 2 and get 256.

$$256x^{2}-72x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 256 for a, -72 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 256\times 32}}{2\times 256}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 256\times 32}}{2\times 256}$$

Multiply -4 times 256.

$$x=\frac{-\left(-72\right)±\sqrt{5184-1024\times 32}}{2\times 256}$$

Multiply -1024 times 32.

$$x=\frac{-\left(-72\right)±\sqrt{5184-32768}}{2\times 256}$$

Add 5184 to -32768.

$$x=\frac{-\left(-72\right)±\sqrt{-27584}}{2\times 256}$$

Take the square root of -27584.

$$x=\frac{-\left(-72\right)±8\sqrt{431}i}{2\times 256}$$

The opposite of -72 is 72.

$$x=\frac{72±8\sqrt{431}i}{2\times 256}$$

Multiply 2 times 256.

$$x=\frac{72±8\sqrt{431}i}{512}$$

Now solve the equation x=\frac{72±8\sqrt{431}i}{512} when ± is plus. Add 72 to 8i\sqrt{431}.

$$x=\frac{72+8\sqrt{431}i}{512}$$

Divide 72+8i\sqrt{431} by 512.

$$x=\frac{9+\sqrt{431}i}{64}$$

Now solve the equation x=\frac{72±8\sqrt{431}i}{512} when ± is minus. Subtract 8i\sqrt{431} from 72.

$$x=\frac{-8\sqrt{431}i+72}{512}$$

Divide 72-8i\sqrt{431} by 512.

$$x=\frac{-\sqrt{431}i+9}{64}$$

The equation is now solved.

$$x=\frac{9+\sqrt{431}i}{64} x=\frac{-\sqrt{431}i+9}{64}$$

Steps for Completing the Square

Expand \left(16x\right)^{2}.

$$16^{2}x^{2}-72x+32=0$$

Calculate 16 to the power of 2 and get 256.

$$256x^{2}-72x+32=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$256x^{2}-72x=-32$$

Divide both sides by 256.

$$\frac{256x^{2}-72x}{256}=-\frac{32}{256}$$

Dividing by 256 undoes the multiplication by 256.

$$x^{2}+\left(-\frac{72}{256}\right)x=-\frac{32}{256}$$

Reduce the fraction \frac{-72}{256} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{9}{32}x=-\frac{32}{256}$$

Reduce the fraction \frac{-32}{256} to lowest terms by extracting and canceling out 32.

$$x^{2}-\frac{9}{32}x=-\frac{1}{8}$$

Divide -\frac{9}{32}, the coefficient of the x term, by 2 to get -\frac{9}{64}. Then add the square of -\frac{9}{64} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{32}x+\left(-\frac{9}{64}\right)^{2}=-\frac{1}{8}+\left(-\frac{9}{64}\right)^{2}$$

Square -\frac{9}{64} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{32}x+\frac{81}{4096}=-\frac{1}{8}+\frac{81}{4096}$$

Add -\frac{1}{8} to \frac{81}{4096} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{32}x+\frac{81}{4096}=-\frac{431}{4096}$$

Factor x^{2}-\frac{9}{32}x+\frac{81}{4096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{64}\right)^{2}=-\frac{431}{4096}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{64}\right)^{2}}=\sqrt{-\frac{431}{4096}}$$

Simplify.

$$x-\frac{9}{64}=\frac{\sqrt{431}i}{64} x-\frac{9}{64}=-\frac{\sqrt{431}i}{64}$$

Add \frac{9}{64} to both sides of the equation.

$$x=\frac{9+\sqrt{431}i}{64} x=\frac{-\sqrt{431}i+9}{64}$$