Solve 0,256x^2+1,024x+0,768=0

$$0,256 x ^ { 2 } + 1,024 x + 0,768 = 0$$

Solve for x

$$x=-3$$

$$x=-1$$

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.

$$0,256x^{2}+1,024x+0,768=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0,256 for a, 1,024 for b, and 0,768 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1,024±\sqrt{1,024^{2}-4\times 0,256\times 0,768}}{2\times 0,256}$$

Square 1,024 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1,024±\sqrt{1,048576-4\times 0,256\times 0,768}}{2\times 0,256}$$

Multiply -4 times 0,256.

$$x=\frac{-1,024±\sqrt{1,048576-1,024\times 0,768}}{2\times 0,256}$$

Multiply -1,024 times 0,768 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1,024±\sqrt{\frac{16384-12288}{15625}}}{2\times 0,256}$$

Add 1,048576 to -0,786432 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1,024±\sqrt{0,262144}}{2\times 0,256}$$

Take the square root of 0,262144.

$$x=\frac{-1,024±\frac{64}{125}}{2\times 0,256}$$

Multiply 2 times 0,256.

$$x=\frac{-1,024±\frac{64}{125}}{0,512}$$

Now solve the equation x=\frac{-1,024±\frac{64}{125}}{0,512} when ± is plus. Add -1,024 to \frac{64}{125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{64}{125}}{0,512}$$

Divide -\frac{64}{125} by 0,512 by multiplying -\frac{64}{125} by the reciprocal of 0,512.

$$x=-1$$

Now solve the equation x=\frac{-1,024±\frac{64}{125}}{0,512} when ± is minus. Subtract \frac{64}{125} from -1,024 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{192}{125}}{0,512}$$

Divide -\frac{192}{125} by 0,512 by multiplying -\frac{192}{125} by the reciprocal of 0,512.

$$x=-3$$

The equation is now solved.

$$x=-1 x=-3$$

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.

$$0,256x^{2}+1,024x+0,768=0$$

Subtract 0,768 from both sides of the equation.

$$0,256x^{2}+1,024x+0,768-0,768=-0,768$$

Subtracting 0,768 from itself leaves 0.

$$0,256x^{2}+1,024x=-0,768$$

Divide both sides of the equation by 0,256, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0,256x^{2}+1,024x}{0,256}=-\frac{0,768}{0,256}$$

Dividing by 0,256 undoes the multiplication by 0,256.

$$x^{2}+\frac{1,024}{0,256}x=-\frac{0,768}{0,256}$$

Divide 1,024 by 0,256 by multiplying 1,024 by the reciprocal of 0,256.

$$x^{2}+4x=-\frac{0,768}{0,256}$$

Divide -0,768 by 0,256 by multiplying -0,768 by the reciprocal of 0,256.

$$x^{2}+4x=-3$$

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

$$x^{2}+4x+2^{2}=-3+2^{2}$$

Square 2.

$$x^{2}+4x+4=-3+4$$

Add -3 to 4.

$$x^{2}+4x+4=1$$

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

$$\left(x+2\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+2=1 x+2=-1$$

Subtract 2 from both sides of the equation.

$$x=-1 x=-3$$