Solve 0.003v^2+0.03v+1.6=10.6

$$0.003 v ^ { 2 } + 0.03 v + 1.6 = 10.6$$

Solve for v

$$v=-60$$

$$v=50$$

Steps Using the Quadratic Formula

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.

$$0.003v^{2}+0.03v+1.6=10.6$$

Subtract 10.6 from both sides of the equation.

$$0.003v^{2}+0.03v+1.6-10.6=10.6-10.6$$

Subtracting 10.6 from itself leaves 0.

$$0.003v^{2}+0.03v+1.6-10.6=0$$

Subtract 10.6 from 1.6 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$0.003v^{2}+0.03v-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.003 for a, 0.03 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-0.03±\sqrt{0.03^{2}-4\times 0.003\left(-9\right)}}{2\times 0.003}$$

Square 0.03 by squaring both the numerator and the denominator of the fraction.

$$v=\frac{-0.03±\sqrt{0.0009-4\times 0.003\left(-9\right)}}{2\times 0.003}$$

Multiply -4 times 0.003.

$$v=\frac{-0.03±\sqrt{0.0009-0.012\left(-9\right)}}{2\times 0.003}$$

Multiply -0.012 times -9.

$$v=\frac{-0.03±\sqrt{0.0009+0.108}}{2\times 0.003}$$

Add 0.0009 to 0.108 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v=\frac{-0.03±\sqrt{0.1089}}{2\times 0.003}$$

Take the square root of 0.1089.

$$v=\frac{-0.03±\frac{33}{100}}{2\times 0.003}$$

Multiply 2 times 0.003.

$$v=\frac{-0.03±\frac{33}{100}}{0.006}$$

Now solve the equation v=\frac{-0.03±\frac{33}{100}}{0.006} when ± is plus. Add -0.03 to \frac{33}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v=\frac{\frac{3}{10}}{0.006}$$

Divide \frac{3}{10} by 0.006 by multiplying \frac{3}{10} by the reciprocal of 0.006.

$$v=50$$

Now solve the equation v=\frac{-0.03±\frac{33}{100}}{0.006} when ± is minus. Subtract \frac{33}{100} from -0.03 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$v=-\frac{\frac{9}{25}}{0.006}$$

Divide -\frac{9}{25} by 0.006 by multiplying -\frac{9}{25} by the reciprocal of 0.006.

$$v=-60$$

The equation is now solved.

$$v=50 v=-60$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$0.003v^{2}+0.03v+1.6=10.6$$

Subtract 1.6 from both sides of the equation.

$$0.003v^{2}+0.03v+1.6-1.6=10.6-1.6$$

Subtracting 1.6 from itself leaves 0.

$$0.003v^{2}+0.03v=10.6-1.6$$

Subtract 1.6 from 10.6 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$0.003v^{2}+0.03v=9$$

Divide both sides of the equation by 0.003, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.003v^{2}+0.03v}{0.003}=\frac{9}{0.003}$$

Dividing by 0.003 undoes the multiplication by 0.003.

$$v^{2}+\frac{0.03}{0.003}v=\frac{9}{0.003}$$

Divide 0.03 by 0.003 by multiplying 0.03 by the reciprocal of 0.003.

$$v^{2}+10v=\frac{9}{0.003}$$

Divide 9 by 0.003 by multiplying 9 by the reciprocal of 0.003.

$$v^{2}+10v=3000$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+10v+5^{2}=3000+5^{2}$$

Square 5.

$$v^{2}+10v+25=3000+25$$

Add 3000 to 25.

$$v^{2}+10v+25=3025$$

Factor v^{2}+10v+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+5\right)^{2}=3025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+5\right)^{2}}=\sqrt{3025}$$

Simplify.

$$v+5=55 v+5=-55$$

Subtract 5 from both sides of the equation.

$$v=50 v=-60$$