Solve -49t^2+102t+100

$$-49 { t }^{ 2 } +102t+100$$

Factor

$$-49\left(t-\frac{51-\sqrt{7501}}{49}\right)\left(t-\frac{\sqrt{7501}+51}{49}\right)$$

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.

$$-49t^{2}+102t+100=0$$

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.

$$t=\frac{-102±\sqrt{102^{2}-4\left(-49\right)\times 100}}{2\left(-49\right)}$$

Square 102.

$$t=\frac{-102±\sqrt{10404-4\left(-49\right)\times 100}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$t=\frac{-102±\sqrt{10404+196\times 100}}{2\left(-49\right)}$$

Multiply 196 times 100.

$$t=\frac{-102±\sqrt{10404+19600}}{2\left(-49\right)}$$

Add 10404 to 19600.

$$t=\frac{-102±\sqrt{30004}}{2\left(-49\right)}$$

Take the square root of 30004.

$$t=\frac{-102±2\sqrt{7501}}{2\left(-49\right)}$$

Multiply 2 times -49.

$$t=\frac{-102±2\sqrt{7501}}{-98}$$

Now solve the equation t=\frac{-102±2\sqrt{7501}}{-98} when ± is plus. Add -102 to 2\sqrt{7501}.

$$t=\frac{2\sqrt{7501}-102}{-98}$$

Divide -102+2\sqrt{7501} by -98.

$$t=\frac{51-\sqrt{7501}}{49}$$

Now solve the equation t=\frac{-102±2\sqrt{7501}}{-98} when ± is minus. Subtract 2\sqrt{7501} from -102.

$$t=\frac{-2\sqrt{7501}-102}{-98}$$

Divide -102-2\sqrt{7501} by -98.

$$t=\frac{\sqrt{7501}+51}{49}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{51-\sqrt{7501}}{49} for x_{1} and \frac{51+\sqrt{7501}}{49} for x_{2}.

$$-49t^{2}+102t+100=-49\left(t-\frac{51-\sqrt{7501}}{49}\right)\left(t-\frac{\sqrt{7501}+51}{49}\right)$$

Evaluate

$$100+102t-49t^{2}$$