Solve 10x^2-40x+1=0

$$10 { x }^{ 2 } -40x+1=0$$

Solve for x

$$x=\frac{\sqrt{390}}{10}+2\approx 3.974841766$$

$$x=-\frac{\sqrt{390}}{10}+2\approx 0.025158234$$

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.

$$10x^{2}-40x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -40 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 10}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-40\right)±\sqrt{1600-40}}{2\times 10}$$

Add 1600 to -40.

$$x=\frac{-\left(-40\right)±\sqrt{1560}}{2\times 10}$$

Take the square root of 1560.

$$x=\frac{-\left(-40\right)±2\sqrt{390}}{2\times 10}$$

The opposite of -40 is 40.

$$x=\frac{40±2\sqrt{390}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{40±2\sqrt{390}}{20}$$

Now solve the equation x=\frac{40±2\sqrt{390}}{20} when ± is plus. Add 40 to 2\sqrt{390}.

$$x=\frac{2\sqrt{390}+40}{20}$$

Divide 40+2\sqrt{390} by 20.

$$x=\frac{\sqrt{390}}{10}+2$$

Now solve the equation x=\frac{40±2\sqrt{390}}{20} when ± is minus. Subtract 2\sqrt{390} from 40.

$$x=\frac{40-2\sqrt{390}}{20}$$

Divide 40-2\sqrt{390} by 20.

$$x=-\frac{\sqrt{390}}{10}+2$$

The equation is now solved.

$$x=\frac{\sqrt{390}}{10}+2 x=-\frac{\sqrt{390}}{10}+2$$

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.

$$10x^{2}-40x+1=0$$

Subtract 1 from both sides of the equation.

$$10x^{2}-40x+1-1=-1$$

Subtracting 1 from itself leaves 0.

$$10x^{2}-40x=-1$$

Divide both sides by 10.

$$\frac{10x^{2}-40x}{10}=-\frac{1}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{40}{10}\right)x=-\frac{1}{10}$$

Divide -40 by 10.

$$x^{2}-4x=-\frac{1}{10}$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{1}{10}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{1}{10}+4$$

Add -\frac{1}{10} to 4.

$$x^{2}-4x+4=\frac{39}{10}$$

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

$$\left(x-2\right)^{2}=\frac{39}{10}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{39}{10}}$$

Simplify.

$$x-2=\frac{\sqrt{390}}{10} x-2=-\frac{\sqrt{390}}{10}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{390}}{10}+2 x=-\frac{\sqrt{390}}{10}+2$$