Find all pairs of real numbers (a, b) such that (x-a)^2+(2x-b)^2=(x-3)^2+(2x)^2

Step-by-step explanation:

Let's expand both sides.

I'm going to use the following identity to expand the binomial squared expressions: or .

Left-hand side:

Reorder so 's are together and that

Right-hand side:

Reorder so 's are together and that

Now let's compare both sides.

If we want both sides to appear exactly the same we need to choose values and such the following are true equations:

So if we solve the system we can find the values and such that the left=right.

Let's solve the first equation for in terms of .

Add on both sides:

Divide both sides by -4:

Reduce (divide top and bottom by -2):

Now let's plug this into second equation:

(I used the identity )

Multiply both sides by 4 to clear the fractions from the problem:

Combine like terms on left hand side:

Subtract 36 on both sides:

Now let's try to factor.

We are going to try to find two numbers that multiply to be 5(-27) and add to be -6.

5(-27)=(5*3)(-9)=15(-9)=-15(9) while -15+9=-6.

So let's replace with and factor by grouping.

This implies or .

Solving the first is easy. Just ad 3 on both sides to get: .

The second requires two steps. Subtract 9 and then divide by 5 on both sides.

.

So let's go back to finding now that we know the values.

If and ,

then .

So one ordered pair that satisfies the equation is:

.

If and ,

then .

Let's multiply top and bottom by 5 to clear the mini-fraction.

So one ordered pair that satisfies the equation is:

.

she drove 134 miles

step-by-step explanation:

the answer is 40.

proof: