Solve 3+12t-4t^2

$$3 + 12 t - 4 t ^ { 2 }$$

Factor

$$-4\left(t-\left(\frac{3}{2}-\sqrt{3}\right)\right)\left(t-\left(\sqrt{3}+\frac{3}{2}\right)\right)$$

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.

$$-4t^{2}+12t+3=0$$

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.

$$t=\frac{-12±\sqrt{12^{2}-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Square 12.

$$t=\frac{-12±\sqrt{144-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$t=\frac{-12±\sqrt{144+16\times 3}}{2\left(-4\right)}$$

Multiply 16 times 3.

$$t=\frac{-12±\sqrt{144+48}}{2\left(-4\right)}$$

Add 144 to 48.

$$t=\frac{-12±\sqrt{192}}{2\left(-4\right)}$$

Take the square root of 192.

$$t=\frac{-12±8\sqrt{3}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$t=\frac{-12±8\sqrt{3}}{-8}$$

Now solve the equation t=\frac{-12±8\sqrt{3}}{-8} when ± is plus. Add -12 to 8\sqrt{3}.

$$t=\frac{8\sqrt{3}-12}{-8}$$

Divide -12+8\sqrt{3} by -8.

$$t=\frac{3}{2}-\sqrt{3}$$

Now solve the equation t=\frac{-12±8\sqrt{3}}{-8} when ± is minus. Subtract 8\sqrt{3} from -12.

$$t=\frac{-8\sqrt{3}-12}{-8}$$

Divide -12-8\sqrt{3} by -8.

$$t=\sqrt{3}+\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{2}-\sqrt{3} for x_{1} and \frac{3}{2}+\sqrt{3} for x_{2}.

$$-4t^{2}+12t+3=-4\left(t-\left(\frac{3}{2}-\sqrt{3}\right)\right)\left(t-\left(\sqrt{3}+\frac{3}{2}\right)\right)$$

Evaluate

$$3+12t-4t^{2}$$