Solve t^2-8t^2+16t=12

$$t ^ { 2 } - 8 t ^ { 2 } + 16 t = 12$$

Solve for t

$$t=\frac{-2\sqrt{5}i+8}{7}\approx 1.142857143-0.638876565i$$

$$t=\frac{8+2\sqrt{5}i}{7}\approx 1.142857143+0.638876565i$$

Steps Using the Quadratic Formula

Combine t^{2} and -8t^{2} to get -7t^{2}.

$$-7t^{2}+16t=12$$

Subtract 12 from both sides.

$$-7t^{2}+16t-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 16 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-16±\sqrt{16^{2}-4\left(-7\right)\left(-12\right)}}{2\left(-7\right)}$$

Square 16.

$$t=\frac{-16±\sqrt{256-4\left(-7\right)\left(-12\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$t=\frac{-16±\sqrt{256+28\left(-12\right)}}{2\left(-7\right)}$$

Multiply 28 times -12.

$$t=\frac{-16±\sqrt{256-336}}{2\left(-7\right)}$$

Add 256 to -336.

$$t=\frac{-16±\sqrt{-80}}{2\left(-7\right)}$$

Take the square root of -80.

$$t=\frac{-16±4\sqrt{5}i}{2\left(-7\right)}$$

Multiply 2 times -7.

$$t=\frac{-16±4\sqrt{5}i}{-14}$$

Now solve the equation t=\frac{-16±4\sqrt{5}i}{-14} when ± is plus. Add -16 to 4i\sqrt{5}.

$$t=\frac{-16+4\sqrt{5}i}{-14}$$

Divide -16+4i\sqrt{5} by -14.

$$t=\frac{-2\sqrt{5}i+8}{7}$$

Now solve the equation t=\frac{-16±4\sqrt{5}i}{-14} when ± is minus. Subtract 4i\sqrt{5} from -16.

$$t=\frac{-4\sqrt{5}i-16}{-14}$$

Divide -16-4i\sqrt{5} by -14.

$$t=\frac{8+2\sqrt{5}i}{7}$$

The equation is now solved.

$$t=\frac{-2\sqrt{5}i+8}{7} t=\frac{8+2\sqrt{5}i}{7}$$

Steps for Completing the Square

Combine t^{2} and -8t^{2} to get -7t^{2}.

$$-7t^{2}+16t=12$$

Divide both sides by -7.

$$\frac{-7t^{2}+16t}{-7}=\frac{12}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$t^{2}+\frac{16}{-7}t=\frac{12}{-7}$$

Divide 16 by -7.

$$t^{2}-\frac{16}{7}t=\frac{12}{-7}$$

Divide 12 by -7.

$$t^{2}-\frac{16}{7}t=-\frac{12}{7}$$

Divide -\frac{16}{7}, the coefficient of the x term, by 2 to get -\frac{8}{7}. Then add the square of -\frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{16}{7}t+\left(-\frac{8}{7}\right)^{2}=-\frac{12}{7}+\left(-\frac{8}{7}\right)^{2}$$

Square -\frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{16}{7}t+\frac{64}{49}=-\frac{12}{7}+\frac{64}{49}$$

Add -\frac{12}{7} to \frac{64}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{16}{7}t+\frac{64}{49}=-\frac{20}{49}$$

Factor t^{2}-\frac{16}{7}t+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{8}{7}\right)^{2}=-\frac{20}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{8}{7}\right)^{2}}=\sqrt{-\frac{20}{49}}$$

Simplify.

$$t-\frac{8}{7}=\frac{2\sqrt{5}i}{7} t-\frac{8}{7}=-\frac{2\sqrt{5}i}{7}$$

Add \frac{8}{7} to both sides of the equation.

$$t=\frac{8+2\sqrt{5}i}{7} t=\frac{-2\sqrt{5}i+8}{7}$$