a=(3+(-8m+13)^1/2)

b=(3-(-8m+13)^1/2)

(a^2+b^2) / (ab) =1

m=?

We are given the following system of equations:

\[
a = \left( 3 + \sqrt{-8m + 13} \right)
\]
\[
b = \left( 3 - \sqrt{-8m + 13} \right)
\]
And the equation:

\[
\frac{a^2 + b^2}{ab} = 1
\]

We need to solve for \( m \).

### Step 1: Simplify \( a^2 + b^2 \)
From the sum of squares identity, we know that:

\[
a^2 + b^2 = (a + b)^2 - 2ab
\]

Let’s calculate \( a + b \) and \( ab \).

#### \( a + b \):
\[
a + b = \left( 3 + \sqrt{-8m + 13} \right) + \left( 3 - \sqrt{-8m + 13} \right)
\]
\[
a + b = 6
\]

#### \( ab \):
\[
ab = \left( 3 + \sqrt{-8m + 13} \right)\left( 3 - \sqrt{-8m + 13} \right)
\]
This is a difference of squares:
\[
ab = 3^2 - \left( \sqrt{-8m + 13} \right)^2
\]
\[
ab = 9 - (-8m + 13)
\]
\[
ab = 9 + 8m - 13
\]
\[
ab = 8m - 4
\]

### Step 2: Substitute into the equation
Now substitute these into the original equation \( \frac{a^2 + b^2}{ab} = 1 \).

First, find \( a^2 + b^2 \):
\[
a^2 + b^2 = (a + b)^2 - 2ab = 6^2 - 2(8m - 4)
\]
\[
a^2 + b^2 = 36 - 2(8m - 4)
\]
\[
a^2 + b^2 = 36 - (16m - 8)
\]
\[
a^2 + b^2 = 44 - 16m
\]

Now substitute into the equation:
\[
\frac{44 - 16m}{8m - 4} = 1
\]

### Step 3: Solve for \( m \)
To solve this equation, multiply both sides by \( 8m - 4 \):
\[
44 - 16m = 8m - 4
\]
Move all terms involving \( m \) to one side and constant terms to the other:
\[
44 + 4 = 8m + 16m
\]
\[
48 = 24m
\]
\[
m = \frac{48}{24} = 2
\]

Thus, the solution is:
\[
m = 2
\]

Let me know if you need more details on any part of this, and here are some related questions to think about:

1. What is the difference of squares formula and how is it used in simplifying products?
2. Why do we need to simplify expressions before solving equations?
3. Can this approach be applied to complex numbers in place of real numbers?
4. What happens when the equation is not directly solvable for \( m \)?
5. How would the solution change if the equation had more complicated terms?

**Tip:** Always check your final solution by substituting it back into the original equation to verify correctness!

a=(3+(-8m+13)^1/2), b=(3-(-8m+13)^1/2), (a^2+b^2)/(ab) = 1, m = ?

Learn how to solve a complex equation involving radicals: a=(3+(-8m+13)^1/2), b=(3-(-8m+13)^1/2), and (a^2+b^2)/(ab) = 1. This problem demonstrates key concepts in algebra, including the sum of squares and difference of squares, and provides a step-by-step solution to find the value of m. Suitable for students in Grades 10-12.

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