Solve x^2+0.491x-1=0

$$x ^ { 2 } + 0.491 x - 1 = 0$$

Solve for x

$$x=\frac{\sqrt{4241081}-491}{2000}\approx 0.784194251$$

$$x=\frac{-\sqrt{4241081}-491}{2000}\approx -1.275194251$$

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.

$$x^{2}+0.491x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0.491 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.491±\sqrt{0.491^{2}-4\left(-1\right)}}{2}$$

Square 0.491 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.491±\sqrt{0.241081-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-0.491±\sqrt{0.241081+4}}{2}$$

Add 0.241081 to 4.

$$x=\frac{-0.491±\sqrt{4.241081}}{2}$$

Take the square root of 4.241081.

$$x=\frac{-0.491±\frac{\sqrt{4241081}}{1000}}{2}$$

Now solve the equation x=\frac{-0.491±\frac{\sqrt{4241081}}{1000}}{2} when ± is plus. Add -0.491 to \frac{\sqrt{4241081}}{1000}.

$$x=\frac{\sqrt{4241081}-491}{2\times 1000}$$

Divide \frac{-491+\sqrt{4241081}}{1000} by 2.

$$x=\frac{\sqrt{4241081}-491}{2000}$$

Now solve the equation x=\frac{-0.491±\frac{\sqrt{4241081}}{1000}}{2} when ± is minus. Subtract \frac{\sqrt{4241081}}{1000} from -0.491.

$$x=\frac{-\sqrt{4241081}-491}{2\times 1000}$$

Divide \frac{-491-\sqrt{4241081}}{1000} by 2.

$$x=\frac{-\sqrt{4241081}-491}{2000}$$

The equation is now solved.

$$x=\frac{\sqrt{4241081}-491}{2000} x=\frac{-\sqrt{4241081}-491}{2000}$$

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.

$$x^{2}+0.491x-1=0$$

Add 1 to both sides of the equation.

$$x^{2}+0.491x-1-\left(-1\right)=-\left(-1\right)$$

Subtracting -1 from itself leaves 0.

$$x^{2}+0.491x=-\left(-1\right)$$

Subtract -1 from 0.

$$x^{2}+0.491x=1$$

Divide 0.491, the coefficient of the x term, by 2 to get 0.2455. Then add the square of 0.2455 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.491x+0.2455^{2}=1+0.2455^{2}$$

Square 0.2455 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.491x+0.06027025=1+0.06027025$$

Add 1 to 0.06027025.

$$x^{2}+0.491x+0.06027025=1.06027025$$

Factor x^{2}+0.491x+0.06027025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.2455\right)^{2}=1.06027025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.2455\right)^{2}}=\sqrt{1.06027025}$$

Simplify.

$$x+0.2455=\frac{\sqrt{4241081}}{2000} x+0.2455=-\frac{\sqrt{4241081}}{2000}$$

Subtract 0.2455 from both sides of the equation.

$$x=\frac{\sqrt{4241081}-491}{2000} x=\frac{-\sqrt{4241081}-491}{2000}$$