Solve Using the Quadratic Formula 2t^2-4t-3=0

$2 t^{2} - 4 t - 3 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 2$ , $b = - 4$ , and $c = - 3$ into the quadratic formula and solve for $t$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (2 \cdot - 3)}}{2 \cdot 2}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$t = \frac{4 \pm \sqrt{16 - 4 \cdot 2 \cdot - 3}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 3$ .

Multiply $- 4$ by $2$ .

$t = \frac{4 \pm \sqrt{16 - 8 \cdot - 3}}{2 \cdot 2}$

Multiply $- 8$ by $- 3$ .

$t = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot 2}$

Add $16$ and $24$ .

$t = \frac{4 \pm \sqrt{40}}{2 \cdot 2}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$t = \frac{4 \pm \sqrt{4 (10)}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$t = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 2}$

Pull terms out from under the radical.

$t = \frac{4 \pm 2 \sqrt{10}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$t = \frac{4 \pm 2 \sqrt{10}}{4}$

Simplify $\frac{4 \pm 2 \sqrt{10}}{4}$ .

$t = \frac{2 \pm \sqrt{10}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{2 \pm \sqrt{10}}{2}$

Decimal Form:

$t = 2.58113883 \dots, - 0.58113883 \dots$