Indices and Surds - Problem Set

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Q1. What is the value of x in the polynomial equation x^3 - 4x^2 + 4x = 0?

Solution:

Factor out x: x(x^2 - 4x + 4) = 0. This gives x = 0 or (x - 2)^2 = 0, so x = 2.

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Q2. What is the solution to the equation 4x + 1 = 2x + 9?

Solution:

Subtract 2x from both sides: 2x + 1 = 9. Then subtract 1: 2x = 8. Finally, divide by 2: x = 4.

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Q3. Which of the following is the correct factorization of x^2 + 5x + 6?

Solution:

Step 1: Find two numbers that multiply to 6 and add to 5: 2 and 3. Step 2: Factor as (x + 2)(x + 3).

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Q4. What is the value of x in the equation 3x^2 + 12x + 12 = 0?

Solution:

Divide the equation by 3: x^2 + 4x + 4 = 0. Factor: (x + 2)(x + 2) = 0, giving x = -2.

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Q5. What is the value of x in the equation x^2 - 5x + 6 = 0?

Solution:

Step 1: Factor the quadratic: (x - 2)(x - 3) = 0. Step 2: Set each factor to zero: x - 2 = 0 or x - 3 = 0. Step 3: Solutions are x = 2 or x = 3.

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Q6. What is the value of x in the equation 3(x - 1) = 2(x + 2)?

Solution:

Step 1: Distribute: 3x - 3 = 2x + 4. Step 2: Subtract 2x from both sides: x - 3 = 4. Step 3: Add 3 to both sides: x = 7.

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Q7. What is the solution to the equation 5(x - 2) = 3x + 4?

Solution:

Distribute: 5x - 10 = 3x + 4. Rearranging gives 2x = 14, so x = 7.

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Q8. What is the solution to the equation 4x^2 - 16 = 0?

Solution:

Step 1: Factor the equation: 4(x^2 - 4) = 0. Step 2: Set x^2 - 4 = 0: (x - 2)(x + 2) = 0. Step 3: Solutions are x = 2 or x = -2.