Solve 3y+1^2-9(y+1)(y-1)=8y-16

$$3 y + 1 ^ { 2 } - 9 ( y + 1 ) ( y - 1 ) = 8 y - 16$$

Solve for y

$$y=-2$$

$$y = \frac{13}{9} = 1\frac{4}{9} \approx 1.444444444$$

Steps Using the Quadratic Formula

Calculate 1 to the power of 2 and get 1.

$$3y+1-9\left(y+1\right)\left(y-1\right)=8y-16$$

Subtract 8y from both sides.

$$3y+1-9\left(y+1\right)\left(y-1\right)-8y=-16$$

Add 16 to both sides.

$$3y+1-9\left(y+1\right)\left(y-1\right)-8y+16=0$$

Use the distributive property to multiply -9 by y+1.

$$3y+1+\left(-9y-9\right)\left(y-1\right)-8y+16=0$$

Use the distributive property to multiply -9y-9 by y-1 and combine like terms.

$$3y+1-9y^{2}+9-8y+16=0$$

Add 1 and 9 to get 10.

$$3y+10-9y^{2}-8y+16=0$$

Combine 3y and -8y to get -5y.

$$-5y+10-9y^{2}+16=0$$

Add 10 and 16 to get 26.

$$-5y+26-9y^{2}=0$$

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.

$$-9y^{2}-5y+26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -5 for b, and 26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-9\right)\times 26}}{2\left(-9\right)}$$

Square -5.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\left(-9\right)\times 26}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$y=\frac{-\left(-5\right)±\sqrt{25+36\times 26}}{2\left(-9\right)}$$

Multiply 36 times 26.

$$y=\frac{-\left(-5\right)±\sqrt{25+936}}{2\left(-9\right)}$$

Add 25 to 936.

$$y=\frac{-\left(-5\right)±\sqrt{961}}{2\left(-9\right)}$$

Take the square root of 961.

$$y=\frac{-\left(-5\right)±31}{2\left(-9\right)}$$

The opposite of -5 is 5.

$$y=\frac{5±31}{2\left(-9\right)}$$

Multiply 2 times -9.

$$y=\frac{5±31}{-18}$$

Now solve the equation y=\frac{5±31}{-18} when ± is plus. Add 5 to 31.

$$y=\frac{36}{-18}$$

Divide 36 by -18.

$$y=-2$$

Now solve the equation y=\frac{5±31}{-18} when ± is minus. Subtract 31 from 5.

$$y=-\frac{26}{-18}$$

Reduce the fraction \frac{-26}{-18} to lowest terms by extracting and canceling out 2.

$$y=\frac{13}{9}$$

The equation is now solved.

$$y=-2 y=\frac{13}{9}$$

Steps for Completing the Square

Calculate 1 to the power of 2 and get 1.

$$3y+1-9\left(y+1\right)\left(y-1\right)=8y-16$$

Subtract 8y from both sides.

$$3y+1-9\left(y+1\right)\left(y-1\right)-8y=-16$$

Use the distributive property to multiply -9 by y+1.

$$3y+1+\left(-9y-9\right)\left(y-1\right)-8y=-16$$

Use the distributive property to multiply -9y-9 by y-1 and combine like terms.

$$3y+1-9y^{2}+9-8y=-16$$

Add 1 and 9 to get 10.

$$3y+10-9y^{2}-8y=-16$$

Combine 3y and -8y to get -5y.

$$-5y+10-9y^{2}=-16$$

Subtract 10 from both sides.

$$-5y-9y^{2}=-16-10$$

Subtract 10 from -16 to get -26.

$$-5y-9y^{2}=-26$$

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.

$$-9y^{2}-5y=-26$$

Divide both sides by -9.

$$\frac{-9y^{2}-5y}{-9}=-\frac{26}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$y^{2}+\left(-\frac{5}{-9}\right)y=-\frac{26}{-9}$$

Divide -5 by -9.

$$y^{2}+\frac{5}{9}y=-\frac{26}{-9}$$

Divide -26 by -9.

$$y^{2}+\frac{5}{9}y=\frac{26}{9}$$

Divide \frac{5}{9}, the coefficient of the x term, by 2 to get \frac{5}{18}. Then add the square of \frac{5}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{5}{9}y+\left(\frac{5}{18}\right)^{2}=\frac{26}{9}+\left(\frac{5}{18}\right)^{2}$$

Square \frac{5}{18} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{5}{9}y+\frac{25}{324}=\frac{26}{9}+\frac{25}{324}$$

Add \frac{26}{9} to \frac{25}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{5}{9}y+\frac{25}{324}=\frac{961}{324}$$

Factor y^{2}+\frac{5}{9}y+\frac{25}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{5}{18}\right)^{2}=\frac{961}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{5}{18}\right)^{2}}=\sqrt{\frac{961}{324}}$$

Simplify.

$$y+\frac{5}{18}=\frac{31}{18} y+\frac{5}{18}=-\frac{31}{18}$$

Subtract \frac{5}{18} from both sides of the equation.

$$y=\frac{13}{9} y=-2$$