Solve -49t^2+55t+3

$$-49 { t }^{ 2 } +55t+3$$

Factor

$$-49\left(t-\frac{55-\sqrt{3613}}{98}\right)\left(t-\frac{\sqrt{3613}+55}{98}\right)$$

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.

$$-49t^{2}+55t+3=0$$

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.

$$t=\frac{-55±\sqrt{55^{2}-4\left(-49\right)\times 3}}{2\left(-49\right)}$$

Square 55.

$$t=\frac{-55±\sqrt{3025-4\left(-49\right)\times 3}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$t=\frac{-55±\sqrt{3025+196\times 3}}{2\left(-49\right)}$$

Multiply 196 times 3.

$$t=\frac{-55±\sqrt{3025+588}}{2\left(-49\right)}$$

Add 3025 to 588.

$$t=\frac{-55±\sqrt{3613}}{2\left(-49\right)}$$

Multiply 2 times -49.

$$t=\frac{-55±\sqrt{3613}}{-98}$$

Now solve the equation t=\frac{-55±\sqrt{3613}}{-98} when ± is plus. Add -55 to \sqrt{3613}.

$$t=\frac{\sqrt{3613}-55}{-98}$$

Divide -55+\sqrt{3613} by -98.

$$t=\frac{55-\sqrt{3613}}{98}$$

Now solve the equation t=\frac{-55±\sqrt{3613}}{-98} when ± is minus. Subtract \sqrt{3613} from -55.

$$t=\frac{-\sqrt{3613}-55}{-98}$$

Divide -55-\sqrt{3613} by -98.

$$t=\frac{\sqrt{3613}+55}{98}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{55-\sqrt{3613}}{98} for x_{1} and \frac{55+\sqrt{3613}}{98} for x_{2}.

$$-49t^{2}+55t+3=-49\left(t-\frac{55-\sqrt{3613}}{98}\right)\left(t-\frac{\sqrt{3613}+55}{98}\right)$$

Evaluate

$$3+55t-49t^{2}$$