Solving for \(a^2 + b^2\) Algebraically
We are given two equations:
- \(a + b = 7\)
- \(ab = 12\)
We need to find the value of \(a^2 + b^2\). We can use the algebraic identity related to the square of a sum:
\( (a+b)^2 = a^2 + 2ab + b^2 \)
Applying the Identity
To find \(a^2 + b^2\), we can rearrange the identity:
\( a^2 + b^2 = (a+b)^2 - 2ab \)
Substitution and Calculation
- Substitute the given values into the rearranged formula:
- \(a+b = 7\)
- \(ab = 12\)
\( a^2 + b^2 = (7)^2 - 2(12) \)
- Calculate the squares and products:
\( (7)^2 = 49 \)
\( 2(12) = 24 \)
- Perform the subtraction:
\( a^2 + b^2 = 49 - 24 \)
\( a^2 + b^2 = 25 \)
Therefore, the value of \(a^2 + b^2\) is 25.
Similar Questions
- For what value of K is the equation \(\frac{0.009}{K} = 0.1\) true?
- Arnab buys 9 oranges and 8 bananas for ₹276. When the cost of an orange is decreased by 10% and that of a banana remains the same, then the cost of 5 oranges and 10 bananas is ₹210. What is the original cost of 4 oranges and 3 bananas?
Reduce to linear form
\(\frac{5x + 1}{2x} = \frac{5}{4}\)
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Important Questions from Algebric Equations
- For what value of K is the equation \(\frac{0.009}{K} = 0.1\) true?
- Arnab buys 9 oranges and 8 bananas for ₹276. When the cost of an orange is decreased by 10% and that of a banana remains the same, then the cost of 5 oranges and 10 bananas is ₹210. What is the original cost of 4 oranges and 3 bananas?
- Find the value of x satisfying: $5(5x^2 - 5) - 5(5x^2 + 7x - 2) = 14$
- The product of two positive numbers is 3645. If the first number is five times the second number, then the sum of the two numbers is:
- The product of two positive numbers is 80. If the first number is five times of the second number, then the sum of the two numbers is:
