Finding a+b using Algebraic Identities

The problem asks us to find the value of \(a+b\), given the values of \(a^2 + b^2\) and \(ab\). We are given:

• \(a^2 + b^2 = 53\)
• \(ab = 14\)

This problem can be solved using a fundamental algebraic identity that relates \((a+b)^2\), \(a^2+b^2\), and \(ab\).

Applying the Algebraic Identity

The relevant algebraic identity is the square of a sum:

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

We can rearrange this identity to group the terms we are given:

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

Now, we can substitute the given values for \(a^2 + b^2\) and \(ab\) into this rearranged identity.

Step-by-Step Calculation

Let's substitute the given values into the equation:

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

Substitute \(a^2 + b^2 = 53\) and \(ab = 14\):

\((a+b)^2 = 53 + 2(14)\)

Now, perform the multiplication:

\((a+b)^2 = 53 + 28\)

Perform the addition:

\((a+b)^2 = 81\)

We have found the value of \((a+b)^2\). To find the value of \(a+b\), we need to take the square root of both sides of the equation.

\(\sqrt{(a+b)^2} = \sqrt{81}\)

\(a+b = \pm 9\)

The equation gives two possible values for \(a+b\): \(+9\) and \(-9\). Since the typical context for such problems and the nature of the options provided usually implies the positive root, we consider the positive value.

Thus, the value of \(a+b\) is 9.

Summary of Steps

Revision Table: Algebraic Identities

Understanding basic algebraic identities is crucial for solving many problems. Here's a quick revision of some important ones:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a+b)(a-b)\)
• \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a+b)\)
• \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a-b)\)
• \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

Additional Information: Solving for a and b Individually

While the problem only asked for \(a+b\), it is sometimes possible to find the individual values of \(a\) and \(b\) from the given equations \(a^2+b^2=53\) and \(ab=14\). We can use the value of \(a+b\) we just found (\(a+b=9\)).

We have a system of two equations:

1. \(a+b = 9\)
2. \(ab = 14\)

From equation (1), we can express \(b\) as \(b = 9-a\). Substitute this into equation (2):

\(a(9-a) = 14\)

\(9a - a^2 = 14\)

Rearrange into a quadratic equation:

\(a^2 - 9a + 14 = 0\)

This quadratic equation can be factored:

\((a-7)(a-2) = 0\)

This gives two possible values for \(a\): \(a=7\) or \(a=2\).

• If \(a=7\), then from \(b=9-a\), \(b = 9-7 = 2\).
• If \(a=2\), then from \(b=9-a\), \(b = 9-2 = 7\).

In both cases, the pair of values for \((a, b)\) is either \((7, 2)\) or \((2, 7)\). Let's check if these pairs satisfy the original conditions:

• If \(a=7, b=2\):
  • \(a+b = 7+2 = 9\) (Matches our result for a+b)
  • \(ab = 7 \times 2 = 14\) (Matches the given condition)
  • \(a^2+b^2 = 7^2 + 2^2 = 49 + 4 = 53\) (Matches the given condition)
• If \(a=2, b=7\):
  • \(a+b = 2+7 = 9\) (Matches our result for a+b)
  • \(ab = 2 \times 7 = 14\) (Matches the given condition)
  • \(a^2+b^2 = 2^2 + 7^2 = 4 + 49 = 53\) (Matches the given condition)

Both pairs satisfy the given conditions, and for both pairs, \(a+b\) is 9. This confirms our result.