Calculating \(a^2 + b^2\) Using Sum and Product
We are given the sum and product of two numbers, \(a\) and \(b\).
- \(a + b = 7\)
- \(ab = 12\)
The goal is to find the value of \(a^2 + b^2\).
Using the Algebraic Identity
Recall the algebraic identity for the square of a binomial sum:
\( (a + b)^2 = a^2 + 2ab + b^2 \)
We can rearrange this identity to solve for \(a^2 + b^2\):
\( a^2 + b^2 = (a + b)^2 - 2ab \)
Substituting Known Values
Substitute the given values \(a + b = 7\) and \(ab = 12\) into the rearranged formula:
\( a^2 + b^2 = (7)^2 - 2(12) \)
Performing the Calculation
First, calculate the square of the sum:
\( 7^2 = 49 \)
Next, calculate twice the product:
\( 2 \times 12 = 24 \)
Finally, subtract twice the product from the square of the sum:
\( a^2 + b^2 = 49 - 24 \)
\( a^2 + b^2 = 25 \)
Thus, the value of \(a^2 + b^2\) is 25.
Similar Questions
- If \(a + b = 12\) and \(ab = 32\), then \(a^2 + b^2 = ?\)
Important Questions from Identities
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
