Solve 58y^2-835.5y+54=0

$$58 { y }^{ 2 } -835.5y+54 = 0$$

Solve for y

$$y = \frac{3 \sqrt{304681} + 1671}{232} \approx 14.34024784$$

$$y=\frac{1671-3\sqrt{304681}}{232}\approx 0.064924574$$

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.

$$58y^{2}-835.5y+54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 58 for a, -835.5 for b, and 54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-835.5\right)±\sqrt{\left(-835.5\right)^{2}-4\times 58\times 54}}{2\times 58}$$

Square -835.5 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-835.5\right)±\sqrt{698060.25-4\times 58\times 54}}{2\times 58}$$

Multiply -4 times 58.

$$y=\frac{-\left(-835.5\right)±\sqrt{698060.25-232\times 54}}{2\times 58}$$

Multiply -232 times 54.

$$y=\frac{-\left(-835.5\right)±\sqrt{698060.25-12528}}{2\times 58}$$

Add 698060.25 to -12528.

$$y=\frac{-\left(-835.5\right)±\sqrt{685532.25}}{2\times 58}$$

Take the square root of 685532.25.

$$y=\frac{-\left(-835.5\right)±\frac{3\sqrt{304681}}{2}}{2\times 58}$$

The opposite of -835.5 is 835.5.

$$y=\frac{835.5±\frac{3\sqrt{304681}}{2}}{2\times 58}$$

Multiply 2 times 58.

$$y=\frac{835.5±\frac{3\sqrt{304681}}{2}}{116}$$

Now solve the equation y=\frac{835.5±\frac{3\sqrt{304681}}{2}}{116} when ± is plus. Add 835.5 to \frac{3\sqrt{304681}}{2}.

$$y=\frac{3\sqrt{304681}+1671}{2\times 116}$$

Divide \frac{1671+3\sqrt{304681}}{2} by 116.

$$y=\frac{3\sqrt{304681}+1671}{232}$$

Now solve the equation y=\frac{835.5±\frac{3\sqrt{304681}}{2}}{116} when ± is minus. Subtract \frac{3\sqrt{304681}}{2} from 835.5.

$$y=\frac{1671-3\sqrt{304681}}{2\times 116}$$

Divide \frac{1671-3\sqrt{304681}}{2} by 116.

$$y=\frac{1671-3\sqrt{304681}}{232}$$

The equation is now solved.

$$y=\frac{3\sqrt{304681}+1671}{232} y=\frac{1671-3\sqrt{304681}}{232}$$

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.

$$58y^{2}-835.5y+54=0$$

Subtract 54 from both sides of the equation.

$$58y^{2}-835.5y+54-54=-54$$

Subtracting 54 from itself leaves 0.

$$58y^{2}-835.5y=-54$$

Divide both sides by 58.

$$\frac{58y^{2}-835.5y}{58}=-\frac{54}{58}$$

Dividing by 58 undoes the multiplication by 58.

$$y^{2}+\left(-\frac{835.5}{58}\right)y=-\frac{54}{58}$$

Divide -835.5 by 58.

$$y^{2}-\frac{1671}{116}y=-\frac{54}{58}$$

Reduce the fraction \frac{-54}{58} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{1671}{116}y=-\frac{27}{29}$$

Divide -\frac{1671}{116}, the coefficient of the x term, by 2 to get -\frac{1671}{232}. Then add the square of -\frac{1671}{232} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{1671}{116}y+\left(-\frac{1671}{232}\right)^{2}=-\frac{27}{29}+\left(-\frac{1671}{232}\right)^{2}$$

Square -\frac{1671}{232} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{1671}{116}y+\frac{2792241}{53824}=-\frac{27}{29}+\frac{2792241}{53824}$$

Add -\frac{27}{29} to \frac{2792241}{53824} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{1671}{116}y+\frac{2792241}{53824}=\frac{2742129}{53824}$$

Factor y^{2}-\frac{1671}{116}y+\frac{2792241}{53824}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{1671}{232}\right)^{2}=\frac{2742129}{53824}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1671}{232}\right)^{2}}=\sqrt{\frac{2742129}{53824}}$$

Simplify.

$$y-\frac{1671}{232}=\frac{3\sqrt{304681}}{232} y-\frac{1671}{232}=-\frac{3\sqrt{304681}}{232}$$

Add \frac{1671}{232} to both sides of the equation.

$$y=\frac{3\sqrt{304681}+1671}{232} y=\frac{1671-3\sqrt{304681}}{232}$$