Solve 1024x^2+768x+1280=0

$$1024 { x }^{ 2 } +768x+1280=0$$

Solve for x (complex solution)

$$x=\frac{-3+\sqrt{71}i}{8}\approx -0.375+1.053268722i$$

$$x=\frac{-\sqrt{71}i-3}{8}\approx -0.375-1.053268722i$$

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.

$$1024x^{2}+768x+1280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1024 for a, 768 for b, and 1280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-768±\sqrt{768^{2}-4\times 1024\times 1280}}{2\times 1024}$$

Square 768.

$$x=\frac{-768±\sqrt{589824-4\times 1024\times 1280}}{2\times 1024}$$

Multiply -4 times 1024.

$$x=\frac{-768±\sqrt{589824-4096\times 1280}}{2\times 1024}$$

Multiply -4096 times 1280.

$$x=\frac{-768±\sqrt{589824-5242880}}{2\times 1024}$$

Add 589824 to -5242880.

$$x=\frac{-768±\sqrt{-4653056}}{2\times 1024}$$

Take the square root of -4653056.

$$x=\frac{-768±256\sqrt{71}i}{2\times 1024}$$

Multiply 2 times 1024.

$$x=\frac{-768±256\sqrt{71}i}{2048}$$

Now solve the equation x=\frac{-768±256\sqrt{71}i}{2048} when ± is plus. Add -768 to 256i\sqrt{71}.

$$x=\frac{-768+256\sqrt{71}i}{2048}$$

Divide -768+256i\sqrt{71} by 2048.

$$x=\frac{-3+\sqrt{71}i}{8}$$

Now solve the equation x=\frac{-768±256\sqrt{71}i}{2048} when ± is minus. Subtract 256i\sqrt{71} from -768.

$$x=\frac{-256\sqrt{71}i-768}{2048}$$

Divide -768-256i\sqrt{71} by 2048.

$$x=\frac{-\sqrt{71}i-3}{8}$$

The equation is now solved.

$$x=\frac{-3+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-3}{8}$$

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.

$$1024x^{2}+768x+1280=0$$

Subtract 1280 from both sides of the equation.

$$1024x^{2}+768x+1280-1280=-1280$$

Subtracting 1280 from itself leaves 0.

$$1024x^{2}+768x=-1280$$

Divide both sides by 1024.

$$\frac{1024x^{2}+768x}{1024}=-\frac{1280}{1024}$$

Dividing by 1024 undoes the multiplication by 1024.

$$x^{2}+\frac{768}{1024}x=-\frac{1280}{1024}$$

Reduce the fraction \frac{768}{1024} to lowest terms by extracting and canceling out 256.

$$x^{2}+\frac{3}{4}x=-\frac{1280}{1024}$$

Reduce the fraction \frac{-1280}{1024} to lowest terms by extracting and canceling out 256.

$$x^{2}+\frac{3}{4}x=-\frac{5}{4}$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{5}{4}+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{5}{4}+\frac{9}{64}$$

Add -\frac{5}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{71}{64}$$

Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{8}\right)^{2}=-\frac{71}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{71}{64}}$$

Simplify.

$$x+\frac{3}{8}=\frac{\sqrt{71}i}{8} x+\frac{3}{8}=-\frac{\sqrt{71}i}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$x=\frac{-3+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-3}{8}$$