Solve -32x^2/130^2+x=264

$$\frac { - 32 x ^ { 2 } } { 130 ^ { 2 } } + x = 264$$

Solve for x (complex solution)

$$x=\frac{4225+65\sqrt{4223}i}{16}\approx 264.0625+263.999992602i$$

$$x=\frac{-65\sqrt{4223}i+4225}{16}\approx 264.0625-263.999992602i$$

Steps Using the Quadratic Formula

Calculate 130 to the power of 2 and get 16900.

$$\frac{-32x^{2}}{16900}+x=264$$

Divide -32x^{2} by 16900 to get -\frac{8}{4225}x^{2}.

$$-\frac{8}{4225}x^{2}+x=264$$

Subtract 264 from both sides.

$$-\frac{8}{4225}x^{2}+x-264=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{8}{4225} for a, 1 for b, and -264 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-\frac{8}{4225}\right)\left(-264\right)}}{2\left(-\frac{8}{4225}\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-\frac{8}{4225}\right)\left(-264\right)}}{2\left(-\frac{8}{4225}\right)}$$

Multiply -4 times -\frac{8}{4225}.

$$x=\frac{-1±\sqrt{1+\frac{32}{4225}\left(-264\right)}}{2\left(-\frac{8}{4225}\right)}$$

Multiply \frac{32}{4225} times -264.

$$x=\frac{-1±\sqrt{1-\frac{8448}{4225}}}{2\left(-\frac{8}{4225}\right)}$$

Add 1 to -\frac{8448}{4225}.

$$x=\frac{-1±\sqrt{-\frac{4223}{4225}}}{2\left(-\frac{8}{4225}\right)}$$

Take the square root of -\frac{4223}{4225}.

$$x=\frac{-1±\frac{\sqrt{4223}i}{65}}{2\left(-\frac{8}{4225}\right)}$$

Multiply 2 times -\frac{8}{4225}.

$$x=\frac{-1±\frac{\sqrt{4223}i}{65}}{-\frac{16}{4225}}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{4223}i}{65}}{-\frac{16}{4225}} when ± is plus. Add -1 to \frac{i\sqrt{4223}}{65}.

$$x=\frac{\frac{\sqrt{4223}i}{65}-1}{-\frac{16}{4225}}$$

Divide -1+\frac{i\sqrt{4223}}{65} by -\frac{16}{4225} by multiplying -1+\frac{i\sqrt{4223}}{65} by the reciprocal of -\frac{16}{4225}.

$$x=\frac{-65\sqrt{4223}i+4225}{16}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{4223}i}{65}}{-\frac{16}{4225}} when ± is minus. Subtract \frac{i\sqrt{4223}}{65} from -1.

$$x=\frac{-\frac{\sqrt{4223}i}{65}-1}{-\frac{16}{4225}}$$

Divide -1-\frac{i\sqrt{4223}}{65} by -\frac{16}{4225} by multiplying -1-\frac{i\sqrt{4223}}{65} by the reciprocal of -\frac{16}{4225}.

$$x=\frac{4225+65\sqrt{4223}i}{16}$$

The equation is now solved.

$$x=\frac{-65\sqrt{4223}i+4225}{16} x=\frac{4225+65\sqrt{4223}i}{16}$$

Steps for Completing the Square

Calculate 130 to the power of 2 and get 16900.

$$\frac{-32x^{2}}{16900}+x=264$$

Divide -32x^{2} by 16900 to get -\frac{8}{4225}x^{2}.

$$-\frac{8}{4225}x^{2}+x=264$$

Divide both sides of the equation by -\frac{8}{4225}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{8}{4225}x^{2}+x}{-\frac{8}{4225}}=\frac{264}{-\frac{8}{4225}}$$

Dividing by -\frac{8}{4225} undoes the multiplication by -\frac{8}{4225}.

$$x^{2}+\frac{1}{-\frac{8}{4225}}x=\frac{264}{-\frac{8}{4225}}$$

Divide 1 by -\frac{8}{4225} by multiplying 1 by the reciprocal of -\frac{8}{4225}.

$$x^{2}-\frac{4225}{8}x=\frac{264}{-\frac{8}{4225}}$$

Divide 264 by -\frac{8}{4225} by multiplying 264 by the reciprocal of -\frac{8}{4225}.

$$x^{2}-\frac{4225}{8}x=-139425$$

Divide -\frac{4225}{8}, the coefficient of the x term, by 2 to get -\frac{4225}{16}. Then add the square of -\frac{4225}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4225}{8}x+\left(-\frac{4225}{16}\right)^{2}=-139425+\left(-\frac{4225}{16}\right)^{2}$$

Square -\frac{4225}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4225}{8}x+\frac{17850625}{256}=-139425+\frac{17850625}{256}$$

Add -139425 to \frac{17850625}{256}.

$$x^{2}-\frac{4225}{8}x+\frac{17850625}{256}=-\frac{17842175}{256}$$

Factor x^{2}-\frac{4225}{8}x+\frac{17850625}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4225}{16}\right)^{2}=-\frac{17842175}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4225}{16}\right)^{2}}=\sqrt{-\frac{17842175}{256}}$$

Simplify.

$$x-\frac{4225}{16}=\frac{65\sqrt{4223}i}{16} x-\frac{4225}{16}=-\frac{65\sqrt{4223}i}{16}$$

Add \frac{4225}{16} to both sides of the equation.

$$x=\frac{4225+65\sqrt{4223}i}{16} x=\frac{-65\sqrt{4223}i+4225}{16}$$