solve the equation 5x^2-25x+30=0

Question

1. solve the equation 5x^2-25x+30=0

 

2. A number minus 6 times it reciprocal is 1. find the number.

Answer 1

1.  First, notice that 5 is a factor of all the terms on the left.  So you can divide both sides by 5:

 

x² - 5x + 6 = 0

 

Now factor the quadratic:

 

(x - 3)(x - 2) = 0

 

When two or more terms multiplied by each other result in 0, that means that at least one of the terms is 0.

 

So x-3 = 0, or x-2 = 0.

 

That means that x = 3 or x = 2.

 

2. Start by calling the "number" x.

 

So according to the statement...

 

"a number" -- x

"minus 6 times its reciprocal"  -- -6(1/x) = -6/x

"is 1" -- equals, or =1

So that gives us an equation:  x - 6/x = 1

 

Now we have to solve it:

 

x - 6/x = 1

 

Multiply both sides by x:

x² - 6 = x

 

Get all the terms together on the left:

x² -x -6 = 0

 

Factor again:

(x - 3)(x + 2) = 0

 

So as before that tells us that, x-3 = 0, or x+2 = 0.  So x = 3 or x = -2