Solve 0.369x^2+52.5x=14

$$0.369 { x }^{ 2 } +52.5x = 14$$

Solve for x

$$x=\frac{10\sqrt{771365}-8750}{123}\approx 0.266168722$$

$$x=\frac{-10\sqrt{771365}-8750}{123}\approx -142.542591486$$

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.

$$0.369x^{2}+52.5x=14$$

Subtract 14 from both sides of the equation.

$$0.369x^{2}+52.5x-14=14-14$$

Subtracting 14 from itself leaves 0.

$$0.369x^{2}+52.5x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.369 for a, 52.5 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-52.5±\sqrt{52.5^{2}-4\times 0.369\left(-14\right)}}{2\times 0.369}$$

Square 52.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-52.5±\sqrt{2756.25-4\times 0.369\left(-14\right)}}{2\times 0.369}$$

Multiply -4 times 0.369.

$$x=\frac{-52.5±\sqrt{2756.25-1.476\left(-14\right)}}{2\times 0.369}$$

Multiply -1.476 times -14.

$$x=\frac{-52.5±\sqrt{2756.25+20.664}}{2\times 0.369}$$

Add 2756.25 to 20.664 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-52.5±\sqrt{2776.914}}{2\times 0.369}$$

Take the square root of 2776.914.

$$x=\frac{-52.5±\frac{3\sqrt{771365}}{50}}{2\times 0.369}$$

Multiply 2 times 0.369.

$$x=\frac{-52.5±\frac{3\sqrt{771365}}{50}}{0.738}$$

Now solve the equation x=\frac{-52.5±\frac{3\sqrt{771365}}{50}}{0.738} when ± is plus. Add -52.5 to \frac{3\sqrt{771365}}{50}.

$$x=\frac{\frac{3\sqrt{771365}}{50}-\frac{105}{2}}{0.738}$$

Divide -\frac{105}{2}+\frac{3\sqrt{771365}}{50} by 0.738 by multiplying -\frac{105}{2}+\frac{3\sqrt{771365}}{50} by the reciprocal of 0.738.

$$x=\frac{10\sqrt{771365}-8750}{123}$$

Now solve the equation x=\frac{-52.5±\frac{3\sqrt{771365}}{50}}{0.738} when ± is minus. Subtract \frac{3\sqrt{771365}}{50} from -52.5.

$$x=\frac{-\frac{3\sqrt{771365}}{50}-\frac{105}{2}}{0.738}$$

Divide -\frac{105}{2}-\frac{3\sqrt{771365}}{50} by 0.738 by multiplying -\frac{105}{2}-\frac{3\sqrt{771365}}{50} by the reciprocal of 0.738.

$$x=\frac{-10\sqrt{771365}-8750}{123}$$

The equation is now solved.

$$x=\frac{10\sqrt{771365}-8750}{123} x=\frac{-10\sqrt{771365}-8750}{123}$$

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.

$$0.369x^{2}+52.5x=14$$

Divide both sides of the equation by 0.369, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.369x^{2}+52.5x}{0.369}=\frac{14}{0.369}$$

Dividing by 0.369 undoes the multiplication by 0.369.

$$x^{2}+\frac{52.5}{0.369}x=\frac{14}{0.369}$$

Divide 52.5 by 0.369 by multiplying 52.5 by the reciprocal of 0.369.

$$x^{2}+\frac{17500}{123}x=\frac{14}{0.369}$$

Divide 14 by 0.369 by multiplying 14 by the reciprocal of 0.369.

$$x^{2}+\frac{17500}{123}x=\frac{14000}{369}$$

Divide \frac{17500}{123}, the coefficient of the x term, by 2 to get \frac{8750}{123}. Then add the square of \frac{8750}{123} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{17500}{123}x+\frac{8750}{123}^{2}=\frac{14000}{369}+\frac{8750}{123}^{2}$$

Square \frac{8750}{123} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{17500}{123}x+\frac{76562500}{15129}=\frac{14000}{369}+\frac{76562500}{15129}$$

Add \frac{14000}{369} to \frac{76562500}{15129} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{17500}{123}x+\frac{76562500}{15129}=\frac{77136500}{15129}$$

Factor x^{2}+\frac{17500}{123}x+\frac{76562500}{15129}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8750}{123}\right)^{2}=\frac{77136500}{15129}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8750}{123}\right)^{2}}=\sqrt{\frac{77136500}{15129}}$$

Simplify.

$$x+\frac{8750}{123}=\frac{10\sqrt{771365}}{123} x+\frac{8750}{123}=-\frac{10\sqrt{771365}}{123}$$

Subtract \frac{8750}{123} from both sides of the equation.

$$x=\frac{10\sqrt{771365}-8750}{123} x=\frac{-10\sqrt{771365}-8750}{123}$$