Solve 5000x^2+15x=16

$$5000 { x }^{ 2 } +15x=16$$

Solve for x

$$x=\frac{\sqrt{12809}-3}{2000}\approx 0.055088426$$

$$x=\frac{-\sqrt{12809}-3}{2000}\approx -0.058088426$$

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.

$$5000x^{2}+15x=16$$

Subtract 16 from both sides of the equation.

$$5000x^{2}+15x-16=16-16$$

Subtracting 16 from itself leaves 0.

$$5000x^{2}+15x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5000 for a, 15 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 5000\left(-16\right)}}{2\times 5000}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 5000\left(-16\right)}}{2\times 5000}$$

Multiply -4 times 5000.

$$x=\frac{-15±\sqrt{225-20000\left(-16\right)}}{2\times 5000}$$

Multiply -20000 times -16.

$$x=\frac{-15±\sqrt{225+320000}}{2\times 5000}$$

Add 225 to 320000.

$$x=\frac{-15±\sqrt{320225}}{2\times 5000}$$

Take the square root of 320225.

$$x=\frac{-15±5\sqrt{12809}}{2\times 5000}$$

Multiply 2 times 5000.

$$x=\frac{-15±5\sqrt{12809}}{10000}$$

Now solve the equation x=\frac{-15±5\sqrt{12809}}{10000} when ± is plus. Add -15 to 5\sqrt{12809}.

$$x=\frac{5\sqrt{12809}-15}{10000}$$

Divide -15+5\sqrt{12809} by 10000.

$$x=\frac{\sqrt{12809}-3}{2000}$$

Now solve the equation x=\frac{-15±5\sqrt{12809}}{10000} when ± is minus. Subtract 5\sqrt{12809} from -15.

$$x=\frac{-5\sqrt{12809}-15}{10000}$$

Divide -15-5\sqrt{12809} by 10000.

$$x=\frac{-\sqrt{12809}-3}{2000}$$

The equation is now solved.

$$x=\frac{\sqrt{12809}-3}{2000} x=\frac{-\sqrt{12809}-3}{2000}$$

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.

$$5000x^{2}+15x=16$$

Divide both sides by 5000.

$$\frac{5000x^{2}+15x}{5000}=\frac{16}{5000}$$

Dividing by 5000 undoes the multiplication by 5000.

$$x^{2}+\frac{15}{5000}x=\frac{16}{5000}$$

Reduce the fraction \frac{15}{5000} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{3}{1000}x=\frac{16}{5000}$$

Reduce the fraction \frac{16}{5000} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{3}{1000}x=\frac{2}{625}$$

Divide \frac{3}{1000}, the coefficient of the x term, by 2 to get \frac{3}{2000}. Then add the square of \frac{3}{2000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{1000}x+\left(\frac{3}{2000}\right)^{2}=\frac{2}{625}+\left(\frac{3}{2000}\right)^{2}$$

Square \frac{3}{2000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{1000}x+\frac{9}{4000000}=\frac{2}{625}+\frac{9}{4000000}$$

Add \frac{2}{625} to \frac{9}{4000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{1000}x+\frac{9}{4000000}=\frac{12809}{4000000}$$

Factor x^{2}+\frac{3}{1000}x+\frac{9}{4000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2000}\right)^{2}=\frac{12809}{4000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2000}\right)^{2}}=\sqrt{\frac{12809}{4000000}}$$

Simplify.

$$x+\frac{3}{2000}=\frac{\sqrt{12809}}{2000} x+\frac{3}{2000}=-\frac{\sqrt{12809}}{2000}$$

Subtract \frac{3}{2000} from both sides of the equation.

$$x=\frac{\sqrt{12809}-3}{2000} x=\frac{-\sqrt{12809}-3}{2000}$$