Solve 30t-t^2-101

$$30 t - t ^ { 2 } - 101$$

Factor

$$-\left(t-\left(15-2\sqrt{31}\right)\right)\left(t-\left(2\sqrt{31}+15\right)\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-t^{2}+30t-101=0$$

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.

$$t=\frac{-30±\sqrt{30^{2}-4\left(-1\right)\left(-101\right)}}{2\left(-1\right)}$$

Square 30.

$$t=\frac{-30±\sqrt{900-4\left(-1\right)\left(-101\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-30±\sqrt{900+4\left(-101\right)}}{2\left(-1\right)}$$

Multiply 4 times -101.

$$t=\frac{-30±\sqrt{900-404}}{2\left(-1\right)}$$

Add 900 to -404.

$$t=\frac{-30±\sqrt{496}}{2\left(-1\right)}$$

Take the square root of 496.

$$t=\frac{-30±4\sqrt{31}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-30±4\sqrt{31}}{-2}$$

Now solve the equation t=\frac{-30±4\sqrt{31}}{-2} when ± is plus. Add -30 to 4\sqrt{31}.

$$t=\frac{4\sqrt{31}-30}{-2}$$

Divide -30+4\sqrt{31} by -2.

$$t=15-2\sqrt{31}$$

Now solve the equation t=\frac{-30±4\sqrt{31}}{-2} when ± is minus. Subtract 4\sqrt{31} from -30.

$$t=\frac{-4\sqrt{31}-30}{-2}$$

Divide -30-4\sqrt{31} by -2.

$$t=2\sqrt{31}+15$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 15-2\sqrt{31} for x_{1} and 15+2\sqrt{31} for x_{2}.

$$-t^{2}+30t-101=-\left(t-\left(15-2\sqrt{31}\right)\right)\left(t-\left(2\sqrt{31}+15\right)\right)$$

Evaluate

$$-t^{2}+30t-101$$