If a + b = 12 and a^2 + b^2 = 70, what is the value of ab? (2019)

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Question: If a + b = 12 and a^2 + b^2 = 70, what is the value of ab? (2019)

Options:

Correct Answer: 20

Exam Year: 2019

Solution:

Using the identity a^2 + b^2 = (a + b)^2 - 2ab, we have 70 = 12^2 - 2ab. Thus, 70 = 144 - 2ab, leading to ab = 37.

If a + b = 12 and a^2 + b^2 = 70, what is the value of ab? (2019)

If a + b = 12 and a^2 + b^2 = 70, what is the value of ab? (2019)

Step 1: Start with the equations given: a + b = 12 and a^2 + b^2 = 70.
Step 2: Recall the identity that relates a^2 + b^2 to a + b and ab: a^2 + b^2 = (a + b)^2 - 2ab.
Step 3: Substitute the value of a + b into the identity: (a + b)^2 = 12^2 = 144.
Step 4: Now rewrite the identity using the values we have: a^2 + b^2 = 144 - 2ab.
Step 5: Set the equation equal to the value of a^2 + b^2: 70 = 144 - 2ab.
Step 6: Rearrange the equation to solve for ab: 2ab = 144 - 70.
Step 7: Calculate 144 - 70, which equals 74: 2ab = 74.
Step 8: Divide both sides by 2 to find ab: ab = 74 / 2 = 37.

Algebraic Identities – Understanding and applying the identity a^2 + b^2 = (a + b)^2 - 2ab to relate sums and products of variables.
System of Equations – Solving for unknowns using given equations involving sums and squares.