Find All Complex Solutions 4t^2-2t-3=-6t

$4 t^{2} - 2 t - 3 = - 6 t$

Move all terms to the left side of the equation and simplify.

Add $6 t$ to both sides of the equation.

$4 t^{2} - 2 t - 3 + 6 t = 0$

Add $- 2 t$ and $6 t$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $4$ .

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

Multiply $- 16$ by $- 3$ .

$t = \frac{- 4 \pm \sqrt{16 + 48}}{2 \cdot 4}$

Add $16$ and $48$ .

$t = \frac{- 4 \pm \sqrt{64}}{2 \cdot 4}$

Rewrite $64$ as $8^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Multiply $2$ by $4$ .

$t = \frac{- 4 \pm 8}{8}$

Simplify $\frac{- 4 \pm 8}{8}$ .

$t = \frac{- 1 \pm 2}{2}$

The final answer is the combination of both solutions.

$t = \frac{1}{2}, - \frac{3}{2}$