Understanding the Algebra Problem

The question asks us to find the value of \(a + b\), given two separate pieces of information about the variables \(a\) and \(b\):

• The difference between \(a\) and \(b\) is 8, expressed as \(a - b = 8\).
• The product of \(a\) and \(b\) is 9, expressed as \(ab = 9\).

We need to use these two pieces of information to determine the value of their sum, \(a + b\).

Using Algebraic Identities to Find a + b

To solve this problem, we can utilize a common algebraic identity that relates the sum, difference, and product of two variables. We know the following identities:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

Notice that both identities contain the terms \(a^2\) and \(b^2\). We can manipulate these identities to find a relationship between \((a + b)^2\), \((a - b)^2\), and \(ab\).

Let's consider the identity for \((a + b)^2\):

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

We can also rearrange the identity for \((a - b)^2\):

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

From this, we can see that \(a^2 + b^2 = (a - b)^2 + 2ab\). Now, substitute this expression for \(a^2 + b^2\) back into the identity for \((a + b)^2\):

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

Simplifying this gives us the useful identity:

\((a + b)^2 = (a - b)^2 + 4ab\)

This identity directly relates the quantities we are given (\(a - b\) and \(ab\)) to the square of the quantity we want to find (\(a + b\)).

Calculating the Value of a + b

Now we can substitute the given values into the identity \((a + b)^2 = (a - b)^2 + 4ab\).

• We are given \(a - b = 8\).
• We are given \(ab = 9\).

Substitute these into the equation:

\((a + b)^2 = (8)^2 + 4(9)\)

Now, perform the calculations:

\((a + b)^2 = 64 + 36\)

\((a + b)^2 = 100\)

To find the value of \(a + b\), we need to take the square root of both sides of the equation:

\(a + b = \pm\sqrt{100}\)

\(a + b = \pm 10\)

Therefore, the value of \(a + b\) can be either \(+10\) or \(-10\).

Conclusion

Using the algebraic identity that connects the sum, difference, and product of two numbers, we found that if \(a - b = 8\) and \(ab = 9\), then \(a + b\) must be \(\pm 10\).

Revision Table: Key Concepts

Additional Information: Verifying the Solution

We can quickly check if there exist numbers \(a\) and \(b\) that satisfy the initial conditions and the result. We have two possible cases for \(a + b\):

Case 1: \(a + b = 10\)

We have the system of equations:

\(a - b = 8\)
\(a + b = 10\)

Adding the two equations gives \(2a = 18\), so \(a = 9\). Substituting \(a = 9\) into \(a + b = 10\) gives \(9 + b = 10\), so \(b = 1\). Let's check the product: \(ab = 9 \times 1 = 9\). This matches the given condition \(ab=9\). So, \(a=9, b=1\) is a valid solution, and \(a+b=10\).

Case 2: \(a + b = -10\)

We have the system of equations:

\(a - b = 8\)
\(a + b = -10\)

Adding the two equations gives \(2a = -2\), so \(a = -1\). Substituting \(a = -1\) into \(a + b = -10\) gives \(-1 + b = -10\), so \(b = -9\). Let's check the product: \(ab = (-1) \times (-9) = 9\). This also matches the given condition \(ab=9\). So, \(a=-1, b=-9\) is another valid solution, and \(a+b=-10\).

Both cases satisfy the original conditions, confirming that \(a + b\) can indeed be \(\pm 10\).