Solve -t^2+4t-280/t^2-4t=0

$$\frac{ - { t }^{ 2 } +4t-280 }{ { t }^{ 2 } -4t } = 0$$

Solve for t

$$t=-2\sqrt{69}i+2\approx 2-16.613247726i$$

$$t=2+2\sqrt{69}i\approx 2+16.613247726i$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by t\left(t-4\right).

$$-t^{2}+4t-280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-280\right)}}{2\left(-1\right)}$$

Square 4.

$$t=\frac{-4±\sqrt{16-4\left(-1\right)\left(-280\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-4±\sqrt{16+4\left(-280\right)}}{2\left(-1\right)}$$

Multiply 4 times -280.

$$t=\frac{-4±\sqrt{16-1120}}{2\left(-1\right)}$$

Add 16 to -1120.

$$t=\frac{-4±\sqrt{-1104}}{2\left(-1\right)}$$

Take the square root of -1104.

$$t=\frac{-4±4\sqrt{69}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-4±4\sqrt{69}i}{-2}$$

Now solve the equation t=\frac{-4±4\sqrt{69}i}{-2} when ± is plus. Add -4 to 4i\sqrt{69}.

$$t=\frac{-4+4\sqrt{69}i}{-2}$$

Divide -4+4i\sqrt{69} by -2.

$$t=-2\sqrt{69}i+2$$

Now solve the equation t=\frac{-4±4\sqrt{69}i}{-2} when ± is minus. Subtract 4i\sqrt{69} from -4.

$$t=\frac{-4\sqrt{69}i-4}{-2}$$

Divide -4-4i\sqrt{69} by -2.

$$t=2+2\sqrt{69}i$$

The equation is now solved.

$$t=-2\sqrt{69}i+2 t=2+2\sqrt{69}i$$

Steps for Completing the Square

Variable t cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by t\left(t-4\right).

$$-t^{2}+4t-280=0$$

Add 280 to both sides. Anything plus zero gives itself.

$$-t^{2}+4t=280$$

Divide both sides by -1.

$$\frac{-t^{2}+4t}{-1}=\frac{280}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{4}{-1}t=\frac{280}{-1}$$

Divide 4 by -1.

$$t^{2}-4t=\frac{280}{-1}$$

Divide 280 by -1.

$$t^{2}-4t=-280$$

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.

$$t^{2}-4t+\left(-2\right)^{2}=-280+\left(-2\right)^{2}$$

Square -2.

$$t^{2}-4t+4=-280+4$$

Add -280 to 4.

$$t^{2}-4t+4=-276$$

Factor t^{2}-4t+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}.

$$\left(t-2\right)^{2}=-276$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-2\right)^{2}}=\sqrt{-276}$$

Simplify.

$$t-2=2\sqrt{69}i t-2=-2\sqrt{69}i$$

Add 2 to both sides of the equation.

$$t=2+2\sqrt{69}i t=-2\sqrt{69}i+2$$