Solve .5t^2-0,4(t+2)=0,7(t-2)

$$.5 t ^ { 2 } - 0,4 ( t + 2 ) = 0,7 ( t - 2 )$$

Solve for t

$$t=1$$

$$t=1.2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -0.4 by t+2.

$$0.5t^{2}-0.4t-0.8=0.7\left(t-2\right)$$

Use the distributive property to multiply 0.7 by t-2.

$$0.5t^{2}-0.4t-0.8=0.7t-1.4$$

Subtract 0.7t from both sides.

$$0.5t^{2}-0.4t-0.8-0.7t=-1.4$$

Combine -0.4t and -0.7t to get -1.1t.

$$0.5t^{2}-1.1t-0.8=-1.4$$

Add 1.4 to both sides.

$$0.5t^{2}-1.1t-0.8+1.4=0$$

Add -0.8 and 1.4 to get 0.6.

$$0.5t^{2}-1.1t+0.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.5 for a, -1.1 for b, and 0.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-1.1\right)±\sqrt{\left(-1.1\right)^{2}-4\times 0.5\times 0.6}}{2\times 0.5}$$

Square -1.1 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-1.1\right)±\sqrt{1.21-4\times 0.5\times 0.6}}{2\times 0.5}$$

Multiply -4 times 0.5.

$$t=\frac{-\left(-1.1\right)±\sqrt{1.21-2\times 0.6}}{2\times 0.5}$$

Multiply -2 times 0.6.

$$t=\frac{-\left(-1.1\right)±\sqrt{1.21-1.2}}{2\times 0.5}$$

Add 1.21 to -1.2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-1.1\right)±\sqrt{0.01}}{2\times 0.5}$$

Take the square root of 0.01.

$$t=\frac{-\left(-1.1\right)±\frac{1}{10}}{2\times 0.5}$$

The opposite of -1.1 is 1.1.

$$t=\frac{1.1±\frac{1}{10}}{2\times 0.5}$$

Multiply 2 times 0.5.

$$t=\frac{1.1±\frac{1}{10}}{1}$$

Now solve the equation t=\frac{1.1±\frac{1}{10}}{1} when ± is plus. Add 1.1 to \frac{1}{10} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{\frac{6}{5}}{1}$$

Divide \frac{6}{5} by 1.

$$t=\frac{6}{5}$$

Now solve the equation t=\frac{1.1±\frac{1}{10}}{1} when ± is minus. Subtract \frac{1}{10} from 1.1 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{1}{1}$$

Divide 1 by 1.

$$t=1$$

The equation is now solved.

$$t=\frac{6}{5} t=1$$

Steps for Completing the Square

Use the distributive property to multiply -0.4 by t+2.

$$0.5t^{2}-0.4t-0.8=0.7\left(t-2\right)$$

Use the distributive property to multiply 0.7 by t-2.

$$0.5t^{2}-0.4t-0.8=0.7t-1.4$$

Subtract 0.7t from both sides.

$$0.5t^{2}-0.4t-0.8-0.7t=-1.4$$

Combine -0.4t and -0.7t to get -1.1t.

$$0.5t^{2}-1.1t-0.8=-1.4$$

Add 0.8 to both sides.

$$0.5t^{2}-1.1t=-1.4+0.8$$

Add -1.4 and 0.8 to get -0.6.

$$0.5t^{2}-1.1t=-0.6$$

Multiply both sides by 2.

$$\frac{0.5t^{2}-1.1t}{0.5}=-\frac{0.6}{0.5}$$

Dividing by 0.5 undoes the multiplication by 0.5.

$$t^{2}+\left(-\frac{1.1}{0.5}\right)t=-\frac{0.6}{0.5}$$

Divide -1.1 by 0.5 by multiplying -1.1 by the reciprocal of 0.5.

$$t^{2}-2.2t=-\frac{0.6}{0.5}$$

Divide -0.6 by 0.5 by multiplying -0.6 by the reciprocal of 0.5.

$$t^{2}-2.2t=-1.2$$

Divide -2.2, the coefficient of the x term, by 2 to get -1.1. Then add the square of -1.1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-2.2t+\left(-1.1\right)^{2}=-1.2+\left(-1.1\right)^{2}$$

Square -1.1 by squaring both the numerator and the denominator of the fraction.

$$t^{2}-2.2t+1.21=-1.2+1.21$$

Add -1.2 to 1.21 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-2.2t+1.21=0.01$$

Factor t^{2}-2.2t+1.21. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-1.1\right)^{2}=0.01$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-1.1\right)^{2}}=\sqrt{0.01}$$

Simplify.

$$t-1.1=\frac{1}{10} t-1.1=-\frac{1}{10}$$

Add 1.1 to both sides of the equation.

$$t=\frac{6}{5} t=1$$