Indices and Surds - Applications

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

Solution:

Factor out x: x(x^2 - 4) = 0. Thus, x = 0 or x^2 - 4 = 0, giving x = 2 or x = -2.

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Q2. Which of the following is a solution to the inequality 3x - 5 < 4?

Solution:

Add 5 to both sides: 3x < 9. Then divide by 3: x < 3.

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Q3. Solve the quadratic equation x^2 - 5x + 6 = 0.

Solution:

Factor the equation: (x - 2)(x - 3) = 0. Thus, x = 2 or x = 3.

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

Solution:

Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Take the square root: x = ±4.

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Q5. Which expression represents the sum of the roots of the equation x^2 - 6x + 8 = 0?

Solution:

Step 1: The sum of the roots is given by -b/a: 6/1 = 6.

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Q6. Which of the following is the result of (x + 2)(x - 2)?

Solution:

Use the difference of squares: (x + 2)(x - 2) = x^2 - 4.

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Q7. What is the solution to the equation 4(x - 1) = 12?

Solution:

Divide both sides by 4: x - 1 = 3. Add 1 to both sides: x = 4.

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Q8. If 5x + 2 > 12, what is the smallest integer value of x?

Solution:

Subtract 2 from both sides: 5x > 10. Then divide by 5: x > 2.

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Q9. What is the solution to the inequality 2x + 1 > 7?

Solution:

Subtract 1 from both sides: 2x > 6. Then divide by 2: x > 3.

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Q10. Which of the following is a solution to the inequality 5x - 7 < 8?

Solution:

Add 7 to both sides: 5x < 15. Divide by 5: x < 3.