Solve 4/4-t=t

$$\frac { 4 } { 4 - t } = t$$

Solve for t

$$t=2$$

Steps Using the Quadratic Formula

Variable t cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by -t+4.

$$4=t\left(-t+4\right)$$

Use the distributive property to multiply t by -t+4.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 4 from both sides.

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

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

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

Square 4.

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

Multiply -4 times -1.

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

Multiply 4 times -4.

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

Add 16 to -16.

$$t=\frac{-4±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

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

Multiply 2 times -1.

$$t=-\frac{4}{-2}$$

Divide -4 by -2.

$$t=2$$

Steps for Completing the Square

Variable t cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by -t+4.

$$4=t\left(-t+4\right)$$

Use the distributive property to multiply t by -t+4.

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

Swap sides so that all variable terms are on the left hand side.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 4 by -1.

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

Divide 4 by -1.

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

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.

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

Square -2.

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

Add -4 to 4.

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

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}.

$$\left(t-2\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-2\right)^{2}}=\sqrt{0}$$

Simplify.

$$t-2=0 t-2=0$$

Add 2 to both sides of the equation.

$$t=2 t=2$$

The equation is now solved. Solutions are the same.

$$t=2$$