Indices and Surds - Case Studies

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Q1. Which of the following represents the equation of a line with a slope of 2 and y-intercept of -3?

Solution:

The slope-intercept form is y = mx + b. Here, m = 2 and b = -3, so y = 2x - 3.

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Q2. Solve for x: 5x + 2 > 12.

Solution:

Step 1: Subtract 2 from both sides: 5x > 10. Step 2: Divide both sides by 5: x > 2.

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Q3. What is the vertex of the quadratic equation y = x^2 - 4x + 3?

Solution:

Step 1: Use the vertex formula x = -b/2a: x = 4/2 = 2. Step 2: Substitute x back into the equation: y = 2^2 - 4(2) + 3 = -1. Vertex is (2, -1).

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Q4. What is the solution to the equation x^2 + 5x + 6 = 0?

Solution:

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

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Q5. What is the value of x in the equation 4x - 7 = 5x + 2?

Solution:

Step 1: Subtract 4x from both sides: -7 = x + 2. Step 2: Subtract 2 from both sides: -9 = x. Therefore, x = -9.

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Q6. What is the value of x in the equation 5(x - 1) = 3x + 7?

Solution:

Step 1: Distribute: 5x - 5 = 3x + 7. Step 2: Subtract 3x from both sides: 2x - 5 = 7. Step 3: Add 5: 2x = 12. Step 4: Divide by 2: x = 6.

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Q7. Solve for x in the equation 4(x - 1) = 2(x + 3).

Solution:

Distribute: 4x - 4 = 2x + 6. Subtract 2x from both sides: 2x - 4 = 6. Add 4: 2x = 10. Divide by 2: x = 5.

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Q8. What is the factored form of the polynomial 2x^2 + 8x?

Solution:

Step 1: Factor out the common term: 2x(x + 4).

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Q9. What is the solution to the inequality 2x + 3 ≥ 7?

Solution:

Step 1: Subtract 3 from both sides: 2x ≥ 4. Step 2: Divide by 2: x ≥ 2.

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Q10. What is the value of x in the equation x^2 - 16 = 0?

Solution:

Factor the equation: (x - 4)(x + 4) = 0. Thus, x = 4 or x = -4.