Solve 37x^2-221x+294=0

$$37 { x }^{ 2 } -221x+294 = 0$$

Solve for x

$$x=2$$

$$x = \frac{147}{37} = 3\frac{36}{37} \approx 3.972972973$$

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.

$$37x^{2}-221x+294=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 37 for a, -221 for b, and 294 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-221\right)±\sqrt{\left(-221\right)^{2}-4\times 37\times 294}}{2\times 37}$$

Square -221.

$$x=\frac{-\left(-221\right)±\sqrt{48841-4\times 37\times 294}}{2\times 37}$$

Multiply -4 times 37.

$$x=\frac{-\left(-221\right)±\sqrt{48841-148\times 294}}{2\times 37}$$

Multiply -148 times 294.

$$x=\frac{-\left(-221\right)±\sqrt{48841-43512}}{2\times 37}$$

Add 48841 to -43512.

$$x=\frac{-\left(-221\right)±\sqrt{5329}}{2\times 37}$$

Take the square root of 5329.

$$x=\frac{-\left(-221\right)±73}{2\times 37}$$

The opposite of -221 is 221.

$$x=\frac{221±73}{2\times 37}$$

Multiply 2 times 37.

$$x=\frac{221±73}{74}$$

Now solve the equation x=\frac{221±73}{74} when ± is plus. Add 221 to 73.

$$x=\frac{294}{74}$$

Reduce the fraction \frac{294}{74} to lowest terms by extracting and canceling out 2.

$$x=\frac{147}{37}$$

Now solve the equation x=\frac{221±73}{74} when ± is minus. Subtract 73 from 221.

$$x=\frac{148}{74}$$

Divide 148 by 74.

$$x=2$$

The equation is now solved.

$$x=\frac{147}{37} x=2$$

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.

$$37x^{2}-221x+294=0$$

Subtract 294 from both sides of the equation.

$$37x^{2}-221x+294-294=-294$$

Subtracting 294 from itself leaves 0.

$$37x^{2}-221x=-294$$

Divide both sides by 37.

$$\frac{37x^{2}-221x}{37}=-\frac{294}{37}$$

Dividing by 37 undoes the multiplication by 37.

$$x^{2}-\frac{221}{37}x=-\frac{294}{37}$$

Divide -\frac{221}{37}, the coefficient of the x term, by 2 to get -\frac{221}{74}. Then add the square of -\frac{221}{74} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{221}{37}x+\left(-\frac{221}{74}\right)^{2}=-\frac{294}{37}+\left(-\frac{221}{74}\right)^{2}$$

Square -\frac{221}{74} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{221}{37}x+\frac{48841}{5476}=-\frac{294}{37}+\frac{48841}{5476}$$

Add -\frac{294}{37} to \frac{48841}{5476} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{221}{37}x+\frac{48841}{5476}=\frac{5329}{5476}$$

Factor x^{2}-\frac{221}{37}x+\frac{48841}{5476}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{221}{74}\right)^{2}=\frac{5329}{5476}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{221}{74}\right)^{2}}=\sqrt{\frac{5329}{5476}}$$

Simplify.

$$x-\frac{221}{74}=\frac{73}{74} x-\frac{221}{74}=-\frac{73}{74}$$

Add \frac{221}{74} to both sides of the equation.

$$x=\frac{147}{37} x=2$$