Solve y^2-2y=5y+60

$$y ^ { 2 } - 2 y = 5 y + 60$$

Solve for y

$$y=-5$$

$$y=12$$

Steps Using Factoring

Subtract 5y from both sides.

$$y^{2}-2y-5y=60$$

Combine -2y and -5y to get -7y.

$$y^{2}-7y=60$$

Subtract 60 from both sides.

$$y^{2}-7y-60=0$$

To solve the equation, factor y^{2}-7y-60 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=-7 ab=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -7.

$$a=-12 b=5$$

Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.

$$\left(y-12\right)\left(y+5\right)$$

To find equation solutions, solve y-12=0 and y+5=0.

$$y=12 y=-5$$

Steps Using Factoring By Grouping

Subtract 5y from both sides.

$$y^{2}-2y-5y=60$$

Combine -2y and -5y to get -7y.

$$y^{2}-7y=60$$

Subtract 60 from both sides.

$$y^{2}-7y-60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-60. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\left(-60\right)=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -7.

$$a=-12 b=5$$

Rewrite y^{2}-7y-60 as \left(y^{2}-12y\right)+\left(5y-60\right).

$$\left(y^{2}-12y\right)+\left(5y-60\right)$$

Factor out y in the first and 5 in the second group.

$$y\left(y-12\right)+5\left(y-12\right)$$

Factor out common term y-12 by using distributive property.

$$\left(y-12\right)\left(y+5\right)$$

To find equation solutions, solve y-12=0 and y+5=0.

$$y=12 y=-5$$

Steps Using the Quadratic Formula

Subtract 5y from both sides.

$$y^{2}-2y-5y=60$$

Combine -2y and -5y to get -7y.

$$y^{2}-7y=60$$

Subtract 60 from both sides.

$$y^{2}-7y-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-60\right)}}{2}$$

Square -7.

$$y=\frac{-\left(-7\right)±\sqrt{49-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$y=\frac{-\left(-7\right)±\sqrt{49+240}}{2}$$

Add 49 to 240.

$$y=\frac{-\left(-7\right)±\sqrt{289}}{2}$$

Take the square root of 289.

$$y=\frac{-\left(-7\right)±17}{2}$$

The opposite of -7 is 7.

$$y=\frac{7±17}{2}$$

Now solve the equation y=\frac{7±17}{2} when ± is plus. Add 7 to 17.

$$y=\frac{24}{2}$$

Divide 24 by 2.

$$y=12$$

Now solve the equation y=\frac{7±17}{2} when ± is minus. Subtract 17 from 7.

$$y=-\frac{10}{2}$$

Divide -10 by 2.

$$y=-5$$

The equation is now solved.

$$y=12 y=-5$$

Steps for Completing the Square

Subtract 5y from both sides.

$$y^{2}-2y-5y=60$$

Combine -2y and -5y to get -7y.

$$y^{2}-7y=60$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-7y+\left(-\frac{7}{2}\right)^{2}=60+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-7y+\frac{49}{4}=60+\frac{49}{4}$$

Add 60 to \frac{49}{4}.

$$y^{2}-7y+\frac{49}{4}=\frac{289}{4}$$

Factor y^{2}-7y+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{7}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{7}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$y-\frac{7}{2}=\frac{17}{2} y-\frac{7}{2}=-\frac{17}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$y=12 y=-5$$