Solve 5.45x^2+0.99x-0.955=0

$$5.45 x ^ { 2 } + 0.99 x - 0.955 = 0$$

Solve for x

$$x=\frac{\sqrt{217991}-99}{1090}\approx 0.337518407$$

$$x=\frac{-\sqrt{217991}-99}{1090}\approx -0.519169783$$

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.

$$5.45x^{2}+0.99x-0.955=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5.45 for a, 0.99 for b, and -0.955 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.99±\sqrt{0.99^{2}-4\times 5.45\left(-0.955\right)}}{2\times 5.45}$$

Square 0.99 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.99±\sqrt{0.9801-4\times 5.45\left(-0.955\right)}}{2\times 5.45}$$

Multiply -4 times 5.45.

$$x=\frac{-0.99±\sqrt{0.9801-21.8\left(-0.955\right)}}{2\times 5.45}$$

Multiply -21.8 times -0.955 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.99±\sqrt{0.9801+20.819}}{2\times 5.45}$$

Add 0.9801 to 20.819 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.99±\sqrt{21.7991}}{2\times 5.45}$$

Take the square root of 21.7991.

$$x=\frac{-0.99±\frac{\sqrt{217991}}{100}}{2\times 5.45}$$

Multiply 2 times 5.45.

$$x=\frac{-0.99±\frac{\sqrt{217991}}{100}}{10.9}$$

Now solve the equation x=\frac{-0.99±\frac{\sqrt{217991}}{100}}{10.9} when ± is plus. Add -0.99 to \frac{\sqrt{217991}}{100}.

$$x=\frac{\sqrt{217991}-99}{10.9\times 100}$$

Divide \frac{-99+\sqrt{217991}}{100} by 10.9 by multiplying \frac{-99+\sqrt{217991}}{100} by the reciprocal of 10.9.

$$x=\frac{\sqrt{217991}-99}{1090}$$

Now solve the equation x=\frac{-0.99±\frac{\sqrt{217991}}{100}}{10.9} when ± is minus. Subtract \frac{\sqrt{217991}}{100} from -0.99.

$$x=\frac{-\sqrt{217991}-99}{10.9\times 100}$$

Divide \frac{-99-\sqrt{217991}}{100} by 10.9 by multiplying \frac{-99-\sqrt{217991}}{100} by the reciprocal of 10.9.

$$x=\frac{-\sqrt{217991}-99}{1090}$$

The equation is now solved.

$$x=\frac{\sqrt{217991}-99}{1090} x=\frac{-\sqrt{217991}-99}{1090}$$

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.

$$5.45x^{2}+0.99x-0.955=0$$

Add 0.955 to both sides of the equation.

$$5.45x^{2}+0.99x-0.955-\left(-0.955\right)=-\left(-0.955\right)$$

Subtracting -0.955 from itself leaves 0.

$$5.45x^{2}+0.99x=-\left(-0.955\right)$$

Subtract -0.955 from 0.

$$5.45x^{2}+0.99x=0.955$$

Divide both sides of the equation by 5.45, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{5.45x^{2}+0.99x}{5.45}=\frac{0.955}{5.45}$$

Dividing by 5.45 undoes the multiplication by 5.45.

$$x^{2}+\frac{0.99}{5.45}x=\frac{0.955}{5.45}$$

Divide 0.99 by 5.45 by multiplying 0.99 by the reciprocal of 5.45.

$$x^{2}+\frac{99}{545}x=\frac{0.955}{5.45}$$

Divide 0.955 by 5.45 by multiplying 0.955 by the reciprocal of 5.45.

$$x^{2}+\frac{99}{545}x=\frac{191}{1090}$$

Divide \frac{99}{545}, the coefficient of the x term, by 2 to get \frac{99}{1090}. Then add the square of \frac{99}{1090} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{99}{545}x+\frac{99}{1090}^{2}=\frac{191}{1090}+\frac{99}{1090}^{2}$$

Square \frac{99}{1090} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{99}{545}x+\frac{9801}{1188100}=\frac{191}{1090}+\frac{9801}{1188100}$$

Add \frac{191}{1090} to \frac{9801}{1188100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{99}{545}x+\frac{9801}{1188100}=\frac{217991}{1188100}$$

Factor x^{2}+\frac{99}{545}x+\frac{9801}{1188100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{99}{1090}\right)^{2}=\frac{217991}{1188100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{99}{1090}\right)^{2}}=\sqrt{\frac{217991}{1188100}}$$

Simplify.

$$x+\frac{99}{1090}=\frac{\sqrt{217991}}{1090} x+\frac{99}{1090}=-\frac{\sqrt{217991}}{1090}$$

Subtract \frac{99}{1090} from both sides of the equation.

$$x=\frac{\sqrt{217991}-99}{1090} x=\frac{-\sqrt{217991}-99}{1090}$$