Solve 5(t^2-5t-16)=0

$$5 ( t ^ { 2 } - 5 t - 16 ) = 0$$

Solve for t

$$t = \frac{\sqrt{89} + 5}{2} \approx 7.216990566$$

$$t=\frac{5-\sqrt{89}}{2}\approx -2.216990566$$

Steps Using the Quadratic Formula

Divide both sides by 5. Zero divided by any non-zero number gives zero.

$$t^{2}-5t-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -5.

$$t=\frac{-\left(-5\right)±\sqrt{25-4\left(-16\right)}}{2}$$

Multiply -4 times -16.

$$t=\frac{-\left(-5\right)±\sqrt{25+64}}{2}$$

Add 25 to 64.

$$t=\frac{-\left(-5\right)±\sqrt{89}}{2}$$

The opposite of -5 is 5.

$$t=\frac{5±\sqrt{89}}{2}$$

Now solve the equation t=\frac{5±\sqrt{89}}{2} when ± is plus. Add 5 to \sqrt{89}.

$$t=\frac{\sqrt{89}+5}{2}$$

Now solve the equation t=\frac{5±\sqrt{89}}{2} when ± is minus. Subtract \sqrt{89} from 5.

$$t=\frac{5-\sqrt{89}}{2}$$

The equation is now solved.

$$t=\frac{\sqrt{89}+5}{2} t=\frac{5-\sqrt{89}}{2}$$

Steps for Completing the Square

Divide both sides by 5. Zero divided by any non-zero number gives zero.

$$t^{2}-5t-16=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$t^{2}-5t=16$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-5t+\left(-\frac{5}{2}\right)^{2}=16+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-5t+\frac{25}{4}=16+\frac{25}{4}$$

Add 16 to \frac{25}{4}.

$$t^{2}-5t+\frac{25}{4}=\frac{89}{4}$$

Factor t^{2}-5t+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{5}{2}\right)^{2}=\frac{89}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{5}{2}\right)^{2}}=\sqrt{\frac{89}{4}}$$

Simplify.

$$t-\frac{5}{2}=\frac{\sqrt{89}}{2} t-\frac{5}{2}=-\frac{\sqrt{89}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$t=\frac{\sqrt{89}+5}{2} t=\frac{5-\sqrt{89}}{2}$$