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

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

Solve for v

$$v=-60$$

$$v=50$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

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

Subtract 10.6 from both sides.

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

Subtract 10.6 from 1.6 to get -9.

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

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.

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

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 1.6 from both sides.

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

Subtract 1.6 from 10.6 to get 9.

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

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.

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

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.

$$v=50 v=-60$$