Solve 10v^2-20v+43=3

$$10 v ^ { 2 } - 20 v + 43 = 3$$

Solve for v

$$v=1+\sqrt{3}i\approx 1+1.732050808i$$

$$v=-\sqrt{3}i+1\approx 1-1.732050808i$$

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.

$$10v^{2}-20v+43=3$$

Subtract 3 from both sides of the equation.

$$10v^{2}-20v+43-3=3-3$$

Subtracting 3 from itself leaves 0.

$$10v^{2}-20v+43-3=0$$

Subtract 3 from 43.

$$10v^{2}-20v+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -20 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 10\times 40}}{2\times 10}$$

Square -20.

$$v=\frac{-\left(-20\right)±\sqrt{400-4\times 10\times 40}}{2\times 10}$$

Multiply -4 times 10.

$$v=\frac{-\left(-20\right)±\sqrt{400-40\times 40}}{2\times 10}$$

Multiply -40 times 40.

$$v=\frac{-\left(-20\right)±\sqrt{400-1600}}{2\times 10}$$

Add 400 to -1600.

$$v=\frac{-\left(-20\right)±\sqrt{-1200}}{2\times 10}$$

Take the square root of -1200.

$$v=\frac{-\left(-20\right)±20\sqrt{3}i}{2\times 10}$$

The opposite of -20 is 20.

$$v=\frac{20±20\sqrt{3}i}{2\times 10}$$

Multiply 2 times 10.

$$v=\frac{20±20\sqrt{3}i}{20}$$

Now solve the equation v=\frac{20±20\sqrt{3}i}{20} when ± is plus. Add 20 to 20i\sqrt{3}.

$$v=\frac{20+20\sqrt{3}i}{20}$$

Divide 20+20i\sqrt{3} by 20.

$$v=1+\sqrt{3}i$$

Now solve the equation v=\frac{20±20\sqrt{3}i}{20} when ± is minus. Subtract 20i\sqrt{3} from 20.

$$v=\frac{-20\sqrt{3}i+20}{20}$$

Divide 20-20i\sqrt{3} by 20.

$$v=-\sqrt{3}i+1$$

The equation is now solved.

$$v=1+\sqrt{3}i v=-\sqrt{3}i+1$$

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.

$$10v^{2}-20v+43=3$$

Subtract 43 from both sides of the equation.

$$10v^{2}-20v+43-43=3-43$$

Subtracting 43 from itself leaves 0.

$$10v^{2}-20v=3-43$$

Subtract 43 from 3.

$$10v^{2}-20v=-40$$

Divide both sides by 10.

$$\frac{10v^{2}-20v}{10}=-\frac{40}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$v^{2}+\left(-\frac{20}{10}\right)v=-\frac{40}{10}$$

Divide -20 by 10.

$$v^{2}-2v=-\frac{40}{10}$$

Divide -40 by 10.

$$v^{2}-2v=-4$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-2v+1=-4+1$$

Add -4 to 1.

$$v^{2}-2v+1=-3$$

Factor v^{2}-2v+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-1\right)^{2}=-3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-1\right)^{2}}=\sqrt{-3}$$

Simplify.

$$v-1=\sqrt{3}i v-1=-\sqrt{3}i$$

Add 1 to both sides of the equation.

$$v=1+\sqrt{3}i v=-\sqrt{3}i+1$$