Solve 289{left({y}_{0}right)}^2+2{y}_{0}+1009=34^281

$$289 { \left( { y }_{ 0 } \right) }^{ 2 } +2 { y }_{ 0 } +1009 = { 34 }^{ 2 } 81$$

Solve for y_0

$$y_{0} = \frac{18 \sqrt{82621} - 1}{289} \approx 17.899295516$$

$$y_{0}=\frac{-18\sqrt{82621}-1}{289}\approx -17.906215931$$

Steps Using the Quadratic Formula

Calculate 34 to the power of 2 and get 1156.

$$289y_{0}^{2}+2y_{0}+1009=1156\times 81$$

Multiply 1156 and 81 to get 93636.

$$289y_{0}^{2}+2y_{0}+1009=93636$$

Subtract 93636 from both sides.

$$289y_{0}^{2}+2y_{0}[phone]=0$$

Subtract 93636 from 1009 to get -92627.

$$289y_{0}^{2}+2y_{0}-92627=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 289 for a, 2 for b, and -92627 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y_{0}=\frac{-2±\sqrt{2^{2}-4\times 289\left(-92627\right)}}{2\times 289}$$

Square 2.

$$y_{0}=\frac{-2±\sqrt{4-4\times 289\left(-92627\right)}}{2\times 289}$$

Multiply -4 times 289.

$$y_{0}=\frac{-2±\sqrt{4-1156\left(-92627\right)}}{2\times 289}$$

Multiply -1156 times -92627.

$$y_{0}=\frac{-2±\sqrt{4+107076812}}{2\times 289}$$

Add 4 to 107076812.

$$y_{0}=\frac{-2±\sqrt{107076816}}{2\times 289}$$

Take the square root of 107076816.

$$y_{0}=\frac{-2±36\sqrt{82621}}{2\times 289}$$

Multiply 2 times 289.

$$y_{0}=\frac{-2±36\sqrt{82621}}{578}$$

Now solve the equation y_{0}=\frac{-2±36\sqrt{82621}}{578} when ± is plus. Add -2 to 36\sqrt{82621}.

$$y_{0}=\frac{36\sqrt{82621}-2}{578}$$

Divide -2+36\sqrt{82621} by 578.

$$y_{0}=\frac{18\sqrt{82621}-1}{289}$$

Now solve the equation y_{0}=\frac{-2±36\sqrt{82621}}{578} when ± is minus. Subtract 36\sqrt{82621} from -2.

$$y_{0}=\frac{-36\sqrt{82621}-2}{578}$$

Divide -2-36\sqrt{82621} by 578.

$$y_{0}=\frac{-18\sqrt{82621}-1}{289}$$

The equation is now solved.

$$y_{0}=\frac{18\sqrt{82621}-1}{289} y_{0}=\frac{-18\sqrt{82621}-1}{289}$$

Steps for Completing the Square

Calculate 34 to the power of 2 and get 1156.

$$289y_{0}^{2}+2y_{0}+1009=1156\times 81$$

Multiply 1156 and 81 to get 93636.

$$289y_{0}^{2}+2y_{0}+1009=93636$$

Subtract 1009 from both sides.

$$289y_{0}^{2}+2y_{0}=93636-1009$$

Subtract 1009 from 93636 to get 92627.

$$289y_{0}^{2}+2y_{0}=92627$$

Divide both sides by 289.

$$\frac{289y_{0}^{2}+2y_{0}}{289}=\frac{92627}{289}$$

Dividing by 289 undoes the multiplication by 289.

$$y_{0}^{2}+\frac{2}{289}y_{0}=\frac{92627}{289}$$

Divide \frac{2}{289}, the coefficient of the x term, by 2 to get \frac{1}{289}. Then add the square of \frac{1}{289} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y_{0}^{2}+\frac{2}{289}y_{0}+\left(\frac{1}{289}\right)^{2}=\frac{92627}{289}+\left(\frac{1}{289}\right)^{2}$$

Square \frac{1}{289} by squaring both the numerator and the denominator of the fraction.

$$y_{0}^{2}+\frac{2}{289}y_{0}+\frac{1}{83521}=\frac{92627}{289}+\frac{1}{83521}$$

Add \frac{92627}{289} to \frac{1}{83521} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y_{0}^{2}+\frac{2}{289}y_{0}+\frac{1}{83521}=\frac{26769204}{83521}$$

Factor y_{0}^{2}+\frac{2}{289}y_{0}+\frac{1}{83521}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y_{0}+\frac{1}{289}\right)^{2}=\frac{26769204}{83521}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y_{0}+\frac{1}{289}\right)^{2}}=\sqrt{\frac{26769204}{83521}}$$

Simplify.

$$y_{0}+\frac{1}{289}=\frac{18\sqrt{82621}}{289} y_{0}+\frac{1}{289}=-\frac{18\sqrt{82621}}{289}$$

Subtract \frac{1}{289} from both sides of the equation.

$$y_{0}=\frac{18\sqrt{82621}-1}{289} y_{0}=\frac{-18\sqrt{82621}-1}{289}$$