Question 26 of 27intermediate🔧 ApplyLong Answer4 marks

What is the algebraic expression describing the following steps—add any two numbers. Multiply this by half of the sum of the two numbers? Prove that this result will be half of the square of the sum of the two numbers.

Correct Answer

Given: Let the two numbers be aa and bb.

Step 1: Sum = a+ba + b

Step 2: Half of the sum = a+b2\frac{a + b}{2}

Step 3: Product of the sum and half the sum = (a+b)×a+b2(a + b) \times \frac{a + b}{2}

Simplification:

(a+b)×a+b2=(a+b)(a+b)2=(a+b)22(a + b) \times \frac{a + b}{2} = \frac{(a + b)(a + b)}{2} = \frac{(a + b)^2}{2}

Answer: The algebraic expression is (a+b)22\frac{(a + b)^2}{2}, which is half of the square of the sum of the two numbers.

Proof: (a+b)×a+b2=(a+b)(a+b)2=(a+b)22(a + b) \times \frac{a + b}{2} = \frac{(a + b)(a + b)}{2} = \frac{(a + b)^2}{2}

∴ This equals half of (a+b)2(a + b)^2, confirming the result is half of the square of the sum.

Exercise: Figure it Out (Section 3) | Q: 8 | (Page Page 21)
For More Understanding

Explanation

The chapter shows how algebraic expressions capture mathematical operations. The steps translate to: (a+b)×a+b2(a + b) \times \frac{a + b}{2}. Simplifying gives (a+b)22\frac{(a + b)^2}{2}, confirming the result is half the square of the sum.

Solution Steps

  1. Let the two numbers be aa and bb

  2. Sum = a+ba + b

  3. Half of sum = a+b2\frac{a + b}{2}

  4. Product = (a+b)×a+b2=(a+b)22(a + b) \times \frac{a + b}{2} = \frac{(a + b)^2}{2}

  5. This is half of (a+b)2(a + b)^2, the square of the sum

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