Binomial Theorem

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Q1. Which of the following represents the binomial expansion of (x + 2)^2?

Solution:

Step 1: Use the formula (a + b)^2 = a^2 + 2ab + b^2. Step 2: Here, a = x and b = 2. Step 3: The expansion is x^2 + 2(2)x + 2^2 = x^2 + 4x + 4.

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Q2. What is the value of x in the equation 4x^2 - 16 = 0?

Solution:

Step 1: Add 16 to both sides: 4x^2 = 16. Step 2: Divide by 4: x^2 = 4. Step 3: Take the square root: x = 2 or x = -2.

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Q3. What is the solution to the inequality 2x + 5 ≥ 3?

Solution:

Subtract 5 from both sides: 2x ≥ -2. Then divide by 2: x ≥ -1.

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Q4. Which expression represents the expansion of (x + 2)^2?

Solution:

Use the formula (a + b)^2 = a^2 + 2ab + b^2: (x + 2)^2 = x^2 + 4x + 4.

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Q5. What is the solution to the equation x^2 + 6x + 9 = 0?

Solution:

Factor the equation: (x + 3)(x + 3) = 0. Thus, x = -3.

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Q6. What is the sum of the roots of the quadratic equation x^2 + 6x + 9 = 0?

Solution:

Step 1: Use the formula -b/a. Here, b = 6 and a = 1. Step 2: The sum of the roots is -6/1 = -6.