Solve 4t^2+8t-5t=0

$$4 { t }^{ 2 } +8t-5t=0$$

Solve for t

$$t=-\frac{3}{4}=-0.75$$

$$t=0$$

Steps Using Factoring

Factor out t.

$$t\left(4t+8-5\right)=0$$

To find equation solutions, solve t=0 and 3+4t=0.

$$t=0 t=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Combine 8t and -5t to get 3t.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-3±\sqrt{3^{2}}}{2\times 4}$$

Take the square root of 3^{2}.

$$t=\frac{-3±3}{2\times 4}$$

Multiply 2 times 4.

$$t=\frac{-3±3}{8}$$

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

$$t=\frac{0}{8}$$

Divide 0 by 8.

$$t=0$$

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

$$t=-\frac{6}{8}$$

Reduce the fraction \frac{-6}{8} to lowest terms by extracting and canceling out 2.

$$t=-\frac{3}{4}$$

The equation is now solved.

$$t=0 t=-\frac{3}{4}$$

Steps for Completing the Square

Combine 8t and -5t to get 3t.

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

Divide both sides by 4.

$$\frac{4t^{2}+3t}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$t^{2}+\frac{3}{4}t=\frac{0}{4}$$

Divide 0 by 4.

$$t^{2}+\frac{3}{4}t=0$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{3}{4}t+\left(\frac{3}{8}\right)^{2}=\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{3}{4}t+\frac{9}{64}=\frac{9}{64}$$

Factor t^{2}+\frac{3}{4}t+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{3}{8}\right)^{2}=\frac{9}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{3}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Simplify.

$$t+\frac{3}{8}=\frac{3}{8} t+\frac{3}{8}=-\frac{3}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$t=0 t=-\frac{3}{4}$$