Solve the following equations

7f+f^2=60
6x^2-x-15=0
6(y-3)^2-(y-3)-15=0

always rearrange things into descending powers. Set equal to zero because after you factor it, you know that the product can be zero only if one of the factors is zero.

f^2+7f-60 = 0
(f+12)(f-5) = 0
f = -12 or 5

6x^2-x-15 = 0
(2x+3)(3x-5) = 0
x = -3/2 or 5/3

6(y-3)^2 - (y-3) - 15 = 0
(2(y-3)+3)(3(y-3)-5) = 0

one way: y-3 = -3/2 or 5/3
That means y = 3/2 or 14/3

other way:
(2y-6+3)(3y-9-14) = 0
(2y-3)(3y-14) = 0
y = 3/2 or 14/3

To solve the given equations, we will use different methods depending on the type of equation.

1) 7f + f^2 = 60:
This is a quadratic equation, so we can solve it by rearranging it into a standard quadratic form and then factorizing or using the quadratic formula.

Step 1: Rearrange the equation:
f^2 + 7f - 60 = 0

Step 2: Factorize the quadratic equation:
(f - 5)(f + 12) = 0

Step 3: Set each factor equal to zero and solve for f:
f - 5 = 0, which gives f = 5
f + 12 = 0, which gives f = -12

So the solutions to the equation 7f + f^2 = 60 are f = 5 and f = -12.

2) 6x^2 - x - 15 = 0:
This is also a quadratic equation, and we can solve it using the quadratic formula.

Step 1: Identify the coefficients:
a = 6, b = -1, and c = -15

Step 2: Apply the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Substituting the values of a, b, and c into the formula:

x = (-(-1) ± sqrt((-1)^2 - 4(6)(-15))) / (2(6))
x = (1 ± sqrt(1 + 360)) / 12
x = (1 ± sqrt(361)) / 12
x = (1 ± 19) / 12

Step 3: Solve for x:
x = (1 + 19) / 12, which gives x = 20/12 = 5/3
x = (1 - 19) / 12, which gives x = -18/12 = -3/2

So the solutions to the equation 6x^2 - x - 15 = 0 are x = 5/3 and x = -3/2.

3) 6(y - 3)^2 - (y - 3) - 15 = 0:
In this equation, we have a quadratic equation in the form of (y - k)^2. We can solve it by substituting a new variable (let's say z) as y - 3, simplifying the equation, and then solving for z.

Step 1: Substitute z for y - 3:
Let z = y - 3.

The equation becomes:
6z^2 - z - 15 = 0

Step 2: Solve the quadratic equation for z:
Using factorization or the quadratic formula, we find:
(z - 3)(6z + 5) = 0

Setting each factor equal to zero:
z - 3 = 0, which gives z = 3
6z + 5 = 0, which gives z = -5/6

Step 3: Substitute back z = y - 3 and solve for y:
For z = 3, y - 3 = 3, which gives y = 6
For z = -5/6, y - 3 = -5/6, which gives y = 17/6

So the solutions to the equation 6(y - 3)^2 - (y - 3) - 15 = 0 are y = 6 and y = 17/6.