Solve 105x^2-40x-4=0

$$105 { x }^{ 2 } -40x-4=0$$

Solve for x

$$x=\frac{2\sqrt{205}}{105}+\frac{4}{21}\approx 0.463196592$$

$$x=-\frac{2\sqrt{205}}{105}+\frac{4}{21}\approx -0.082244211$$

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.

$$105x^{2}-40x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 105 for a, -40 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 105\left(-4\right)}}{2\times 105}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 105\left(-4\right)}}{2\times 105}$$

Multiply -4 times 105.

$$x=\frac{-\left(-40\right)±\sqrt{1600-420\left(-4\right)}}{2\times 105}$$

Multiply -420 times -4.

$$x=\frac{-\left(-40\right)±\sqrt{1600+1680}}{2\times 105}$$

Add 1600 to 1680.

$$x=\frac{-\left(-40\right)±\sqrt{3280}}{2\times 105}$$

Take the square root of 3280.

$$x=\frac{-\left(-40\right)±4\sqrt{205}}{2\times 105}$$

The opposite of -40 is 40.

$$x=\frac{40±4\sqrt{205}}{2\times 105}$$

Multiply 2 times 105.

$$x=\frac{40±4\sqrt{205}}{210}$$

Now solve the equation x=\frac{40±4\sqrt{205}}{210} when ± is plus. Add 40 to 4\sqrt{205}.

$$x=\frac{4\sqrt{205}+40}{210}$$

Divide 40+4\sqrt{205} by 210.

$$x=\frac{2\sqrt{205}}{105}+\frac{4}{21}$$

Now solve the equation x=\frac{40±4\sqrt{205}}{210} when ± is minus. Subtract 4\sqrt{205} from 40.

$$x=\frac{40-4\sqrt{205}}{210}$$

Divide 40-4\sqrt{205} by 210.

$$x=-\frac{2\sqrt{205}}{105}+\frac{4}{21}$$

The equation is now solved.

$$x=\frac{2\sqrt{205}}{105}+\frac{4}{21} x=-\frac{2\sqrt{205}}{105}+\frac{4}{21}$$

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.

$$105x^{2}-40x-4=0$$

Add 4 to both sides of the equation.

$$105x^{2}-40x-4-\left(-4\right)=-\left(-4\right)$$

Subtracting -4 from itself leaves 0.

$$105x^{2}-40x=-\left(-4\right)$$

Subtract -4 from 0.

$$105x^{2}-40x=4$$

Divide both sides by 105.

$$\frac{105x^{2}-40x}{105}=\frac{4}{105}$$

Dividing by 105 undoes the multiplication by 105.

$$x^{2}+\left(-\frac{40}{105}\right)x=\frac{4}{105}$$

Reduce the fraction \frac{-40}{105} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{8}{21}x=\frac{4}{105}$$

Divide -\frac{8}{21}, the coefficient of the x term, by 2 to get -\frac{4}{21}. Then add the square of -\frac{4}{21} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{21}x+\left(-\frac{4}{21}\right)^{2}=\frac{4}{105}+\left(-\frac{4}{21}\right)^{2}$$

Square -\frac{4}{21} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{21}x+\frac{16}{441}=\frac{4}{105}+\frac{16}{441}$$

Add \frac{4}{105} to \frac{16}{441} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{21}x+\frac{16}{441}=\frac{164}{2205}$$

Factor x^{2}-\frac{8}{21}x+\frac{16}{441}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{21}\right)^{2}=\frac{164}{2205}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{21}\right)^{2}}=\sqrt{\frac{164}{2205}}$$

Simplify.

$$x-\frac{4}{21}=\frac{2\sqrt{205}}{105} x-\frac{4}{21}=-\frac{2\sqrt{205}}{105}$$

Add \frac{4}{21} to both sides of the equation.

$$x=\frac{2\sqrt{205}}{105}+\frac{4}{21} x=-\frac{2\sqrt{205}}{105}+\frac{4}{21}$$