Solve .0273x^2-.3687x-42.1541=0

$$.0273 { x }^{ 2 } -.3687x-42.1541=0$$

Solve for x

$$x=\frac{\sqrt{473916741}}{546}+\frac{1229}{182}\approx 46.623862393$$

$$x=-\frac{\sqrt{473916741}}{546}+\frac{1229}{182}\approx -33.118367888$$

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.

$$0.0273x^{2}-0.3687x-42.1541=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0273 for a, -0.3687 for b, and -42.1541 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.3687\right)±\sqrt{\left(-0.3687\right)^{2}-4\times 0.0273\left(-42.1541\right)}}{2\times 0.0273}$$

Square -0.3687 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.3687\right)±\sqrt{0.13593969-4\times 0.0273\left(-42.1541\right)}}{2\times 0.0273}$$

Multiply -4 times 0.0273.

$$x=\frac{-\left(-0.3687\right)±\sqrt{0.13593969-0.1092\left(-42.1541\right)}}{2\times 0.0273}$$

Multiply -0.1092 times -42.1541 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.3687\right)±\sqrt{0.13593969+4.60322772}}{2\times 0.0273}$$

Add 0.13593969 to 4.60322772 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.3687\right)±\sqrt{4.73916741}}{2\times 0.0273}$$

Take the square root of 4.73916741.

$$x=\frac{-\left(-0.3687\right)±\frac{\sqrt{473916741}}{10000}}{2\times 0.0273}$$

The opposite of -0.3687 is 0.3687.

$$x=\frac{0.3687±\frac{\sqrt{473916741}}{10000}}{2\times 0.0273}$$

Multiply 2 times 0.0273.

$$x=\frac{0.3687±\frac{\sqrt{473916741}}{10000}}{0.0546}$$

Now solve the equation x=\frac{0.3687±\frac{\sqrt{473916741}}{10000}}{0.0546} when ± is plus. Add 0.3687 to \frac{\sqrt{473916741}}{10000}.

$$x=\frac{\sqrt{473916741}+3687}{0.0546\times 10000}$$

Divide \frac{3687+\sqrt{473916741}}{10000} by 0.0546 by multiplying \frac{3687+\sqrt{473916741}}{10000} by the reciprocal of 0.0546.

$$x=\frac{\sqrt{473916741}}{546}+\frac{1229}{182}$$

Now solve the equation x=\frac{0.3687±\frac{\sqrt{473916741}}{10000}}{0.0546} when ± is minus. Subtract \frac{\sqrt{473916741}}{10000} from 0.3687.

$$x=\frac{3687-\sqrt{473916741}}{0.0546\times 10000}$$

Divide \frac{3687-\sqrt{473916741}}{10000} by 0.0546 by multiplying \frac{3687-\sqrt{473916741}}{10000} by the reciprocal of 0.0546.

$$x=-\frac{\sqrt{473916741}}{546}+\frac{1229}{182}$$

The equation is now solved.

$$x=\frac{\sqrt{473916741}}{546}+\frac{1229}{182} x=-\frac{\sqrt{473916741}}{546}+\frac{1229}{182}$$

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.

$$0.0273x^{2}-0.3687x-42.1541=0$$

Add 42.1541 to both sides of the equation.

$$0.0273x^{2}-0.3687x-42.1541-\left(-42.1541\right)=-\left(-42.1541\right)$$

Subtracting -42.1541 from itself leaves 0.

$$0.0273x^{2}-0.3687x=-\left(-42.1541\right)$$

Subtract -42.1541 from 0.

$$0.0273x^{2}-0.3687x=42.1541$$

Divide both sides of the equation by 0.0273, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0273x^{2}-0.3687x}{0.0273}=\frac{42.1541}{0.0273}$$

Dividing by 0.0273 undoes the multiplication by 0.0273.

$$x^{2}+\left(-\frac{0.3687}{0.0273}\right)x=\frac{42.1541}{0.0273}$$

Divide -0.3687 by 0.0273 by multiplying -0.3687 by the reciprocal of 0.0273.

$$x^{2}-\frac{1229}{91}x=\frac{42.1541}{0.0273}$$

Divide 42.1541 by 0.0273 by multiplying 42.1541 by the reciprocal of 0.0273.

$$x^{2}-\frac{1229}{91}x=\frac{421541}{273}$$

Divide -\frac{1229}{91}, the coefficient of the x term, by 2 to get -\frac{1229}{182}. Then add the square of -\frac{1229}{182} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1229}{91}x+\left(-\frac{1229}{182}\right)^{2}=\frac{421541}{273}+\left(-\frac{1229}{182}\right)^{2}$$

Square -\frac{1229}{182} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1229}{91}x+\frac{1510441}{33124}=\frac{421541}{273}+\frac{1510441}{33124}$$

Add \frac{421541}{273} to \frac{1510441}{33124} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1229}{91}x+\frac{1510441}{33124}=\frac{157972247}{99372}$$

Factor x^{2}-\frac{1229}{91}x+\frac{1510441}{33124}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1229}{182}\right)^{2}=\frac{157972247}{99372}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1229}{182}\right)^{2}}=\sqrt{\frac{157972247}{99372}}$$

Simplify.

$$x-\frac{1229}{182}=\frac{\sqrt{473916741}}{546} x-\frac{1229}{182}=-\frac{\sqrt{473916741}}{546}$$

Add \frac{1229}{182} to both sides of the equation.

$$x=\frac{\sqrt{473916741}}{546}+\frac{1229}{182} x=-\frac{\sqrt{473916741}}{546}+\frac{1229}{182}$$