Solve 87x^2-365x+250=0

$$87 { x }^{ 2 } -365x+250 = 0$$

Solve for x

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

$$x=\frac{25}{29}\approx 0.862068966$$

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.

$$87x^{2}-365x+250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 87 for a, -365 for b, and 250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-365\right)±\sqrt{\left(-365\right)^{2}-4\times 87\times 250}}{2\times 87}$$

Square -365.

$$x=\frac{-\left(-365\right)±\sqrt{133225-4\times 87\times 250}}{2\times 87}$$

Multiply -4 times 87.

$$x=\frac{-\left(-365\right)±\sqrt{133225-348\times 250}}{2\times 87}$$

Multiply -348 times 250.

$$x=\frac{-\left(-365\right)±\sqrt{133225-87000}}{2\times 87}$$

Add 133225 to -87000.

$$x=\frac{-\left(-365\right)±\sqrt{46225}}{2\times 87}$$

Take the square root of 46225.

$$x=\frac{-\left(-365\right)±215}{2\times 87}$$

The opposite of -365 is 365.

$$x=\frac{365±215}{2\times 87}$$

Multiply 2 times 87.

$$x=\frac{365±215}{174}$$

Now solve the equation x=\frac{365±215}{174} when ± is plus. Add 365 to 215.

$$x=\frac{580}{174}$$

Reduce the fraction \frac{580}{174} to lowest terms by extracting and canceling out 58.

$$x=\frac{10}{3}$$

Now solve the equation x=\frac{365±215}{174} when ± is minus. Subtract 215 from 365.

$$x=\frac{150}{174}$$

Reduce the fraction \frac{150}{174} to lowest terms by extracting and canceling out 6.

$$x=\frac{25}{29}$$

The equation is now solved.

$$x=\frac{10}{3} x=\frac{25}{29}$$

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.

$$87x^{2}-365x+250=0$$

Subtract 250 from both sides of the equation.

$$87x^{2}-365x+250-250=-250$$

Subtracting 250 from itself leaves 0.

$$87x^{2}-365x=-250$$

Divide both sides by 87.

$$\frac{87x^{2}-365x}{87}=-\frac{250}{87}$$

Dividing by 87 undoes the multiplication by 87.

$$x^{2}-\frac{365}{87}x=-\frac{250}{87}$$

Divide -\frac{365}{87}, the coefficient of the x term, by 2 to get -\frac{365}{174}. Then add the square of -\frac{365}{174} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{365}{87}x+\left(-\frac{365}{174}\right)^{2}=-\frac{250}{87}+\left(-\frac{365}{174}\right)^{2}$$

Square -\frac{365}{174} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{365}{87}x+\frac{133225}{30276}=-\frac{250}{87}+\frac{133225}{30276}$$

Add -\frac{250}{87} to \frac{133225}{30276} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{365}{87}x+\frac{133225}{30276}=\frac{46225}{30276}$$

Factor x^{2}-\frac{365}{87}x+\frac{133225}{30276}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{365}{174}\right)^{2}=\frac{46225}{30276}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{365}{174}\right)^{2}}=\sqrt{\frac{46225}{30276}}$$

Simplify.

$$x-\frac{365}{174}=\frac{215}{174} x-\frac{365}{174}=-\frac{215}{174}$$

Add \frac{365}{174} to both sides of the equation.

$$x=\frac{10}{3} x=\frac{25}{29}$$