Calculate a² + b² Value

This problem asks for the value of $a^2 + b^2$ given the sum ($a+b$) and product ($ab$) of two numbers. We are provided with:

• $a + b = 7$
• $ab = 3$

Our goal is to find the value of $a^2 + b^2$.

Using the Relevant Algebraic Identity

We can solve this problem by using the algebraic identity for the square of a binomial sum:

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

To find $a^2 + b^2$, we can rearrange this formula. By subtracting $2ab$ from both sides, we get:

$$ a^2 + b^2 = (a+b)^2 - 2ab $$

Substituting the Given Values

Now, we substitute the known values of $a+b=7$ and $ab=3$ into the rearranged formula:

$$ a^2 + b^2 = (7)^2 - 2(3) $$

Step-by-Step Calculation

The calculation proceeds as follows:

1. Calculate the square of the sum: $7^2 = 7 \times 7 = 49$.
2. Calculate twice the product: $2 \times 3 = 6$.
3. Subtract the result from step 2 from the result in step 1: $49 - 6 = 43$.

So, the value of $a^2 + b^2$ is 43.

$$ a^2 + b^2 = 43 $$