Solve left(2xright)^2-14x+32=0

$${ \left(2x \right) }^{ 2 } -14x+32=0$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{79}i}{4}\approx 1.75+2.222048604i$$

$$x=\frac{-\sqrt{79}i+7}{4}\approx 1.75-2.222048604i$$

Steps Using the Quadratic Formula

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

$$2^{2}x^{2}-14x+32=0$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-14x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -14 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\times 32}}{2\times 4}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 4\times 32}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-14\right)±\sqrt{196-16\times 32}}{2\times 4}$$

Multiply -16 times 32.

$$x=\frac{-\left(-14\right)±\sqrt{196-512}}{2\times 4}$$

Add 196 to -512.

$$x=\frac{-\left(-14\right)±\sqrt{-316}}{2\times 4}$$

Take the square root of -316.

$$x=\frac{-\left(-14\right)±2\sqrt{79}i}{2\times 4}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{79}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{14±2\sqrt{79}i}{8}$$

Now solve the equation x=\frac{14±2\sqrt{79}i}{8} when ± is plus. Add 14 to 2i\sqrt{79}.

$$x=\frac{14+2\sqrt{79}i}{8}$$

Divide 14+2i\sqrt{79} by 8.

$$x=\frac{7+\sqrt{79}i}{4}$$

Now solve the equation x=\frac{14±2\sqrt{79}i}{8} when ± is minus. Subtract 2i\sqrt{79} from 14.

$$x=\frac{-2\sqrt{79}i+14}{8}$$

Divide 14-2i\sqrt{79} by 8.

$$x=\frac{-\sqrt{79}i+7}{4}$$

The equation is now solved.

$$x=\frac{7+\sqrt{79}i}{4} x=\frac{-\sqrt{79}i+7}{4}$$

Steps for Completing the Square

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

$$2^{2}x^{2}-14x+32=0$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-14x+32=0$$

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

$$4x^{2}-14x=-32$$

Divide both sides by 4.

$$\frac{4x^{2}-14x}{4}=-\frac{32}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{14}{4}\right)x=-\frac{32}{4}$$

Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{2}x=-\frac{32}{4}$$

Divide -32 by 4.

$$x^{2}-\frac{7}{2}x=-8$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-8+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-8+\frac{49}{16}$$

Add -8 to \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{79}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=-\frac{79}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{-\frac{79}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{\sqrt{79}i}{4} x-\frac{7}{4}=-\frac{\sqrt{79}i}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=\frac{7+\sqrt{79}i}{4} x=\frac{-\sqrt{79}i+7}{4}$$