Solve 16x^2-128x+319=0

$$16 { x }^{ 2 } -128x+319=0$$

Solve for x (complex solution)

$$x=\frac{3\sqrt{7}i}{4}+4\approx 4+1.984313483i$$

$$x=-\frac{3\sqrt{7}i}{4}+4\approx 4-1.984313483i$$

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.

$$16x^{2}-128x+319=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -128 for b, and 319 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-128\right)±\sqrt{\left(-128\right)^{2}-4\times 16\times 319}}{2\times 16}$$

Square -128.

$$x=\frac{-\left(-128\right)±\sqrt{16384-4\times 16\times 319}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-128\right)±\sqrt{16384-64\times 319}}{2\times 16}$$

Multiply -64 times 319.

$$x=\frac{-\left(-128\right)±\sqrt{16384-20416}}{2\times 16}$$

Add 16384 to -20416.

$$x=\frac{-\left(-128\right)±\sqrt{-4032}}{2\times 16}$$

Take the square root of -4032.

$$x=\frac{-\left(-128\right)±24\sqrt{7}i}{2\times 16}$$

The opposite of -128 is 128.

$$x=\frac{128±24\sqrt{7}i}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{128±24\sqrt{7}i}{32}$$

Now solve the equation x=\frac{128±24\sqrt{7}i}{32} when ± is plus. Add 128 to 24i\sqrt{7}.

$$x=\frac{128+24\sqrt{7}i}{32}$$

Divide 128+24i\sqrt{7} by 32.

$$x=\frac{3\sqrt{7}i}{4}+4$$

Now solve the equation x=\frac{128±24\sqrt{7}i}{32} when ± is minus. Subtract 24i\sqrt{7} from 128.

$$x=\frac{-24\sqrt{7}i+128}{32}$$

Divide 128-24i\sqrt{7} by 32.

$$x=-\frac{3\sqrt{7}i}{4}+4$$

The equation is now solved.

$$x=\frac{3\sqrt{7}i}{4}+4 x=-\frac{3\sqrt{7}i}{4}+4$$

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.

$$16x^{2}-128x+319=0$$

Subtract 319 from both sides of the equation.

$$16x^{2}-128x+319-319=-319$$

Subtracting 319 from itself leaves 0.

$$16x^{2}-128x=-319$$

Divide both sides by 16.

$$\frac{16x^{2}-128x}{16}=-\frac{319}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{128}{16}\right)x=-\frac{319}{16}$$

Divide -128 by 16.

$$x^{2}-8x=-\frac{319}{16}$$

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

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{319}{16}+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-\frac{319}{16}+16$$

Add -\frac{319}{16} to 16.

$$x^{2}-8x+16=-\frac{63}{16}$$

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

$$\left(x-4\right)^{2}=-\frac{63}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-\frac{63}{16}}$$

Simplify.

$$x-4=\frac{3\sqrt{7}i}{4} x-4=-\frac{3\sqrt{7}i}{4}$$

Add 4 to both sides of the equation.

$$x=\frac{3\sqrt{7}i}{4}+4 x=-\frac{3\sqrt{7}i}{4}+4$$