Quadratic Inequalities Practice Worksheet
Mastering quadratic inequalities requires a solid grasp of both algebraic factorisation and the geometric properties of parabolas. This worksheet is designed to help you practice identifying critical values and determining the correct solution intervals.
If you need a refresher on the step-by-step methods used to find these regions, check out our comprehensive guide on How to Solve Quadratic Inequalities.
Practice Questions & Solutions
Solve x² - 6x + 8 > 0
1. Find Critical Values: x² - 6x + 8 = 0 factors to (x - 2)(x - 4) = 0. Values are x = 2 and x = 4.
2. Sketch: A positive parabola crossing at 2 and 4.
3. Identify Region: We want > 0 (above the x-axis). This occurs in the "tails."
Final Answer: x < 2 or x > 4
Solve x² + 2x + 1 ≥ 0
1. Find Critical Values: x² + 2x + 1 = 0 factors to (x + 1)² = 0. The only value is x = -1.
2. Sketch: The parabola touches the x-axis at -1 and opens upwards.
3. Identify Region: Since a squared number is always positive or zero, the expression is ≥ 0 for all real numbers.
Final Answer: All real numbers (x ∈ ℝ)
Solve x² + x + 1 > 0
1. Find Critical Values: The discriminant (b² - 4ac) is 1 - 4 = -3. There are no real roots.
2. Sketch: The parabola is entirely above the x-axis.
3. Identify Region: We want > 0. Since the curve never touches or crosses the axis, it is always greater than zero.
Final Answer: All real numbers (x ∈ ℝ)
Solve 7x² + 21x - 28 < 0
1. Simplify: Divide by 7 to get x² + 3x - 4 < 0.
2. Find Critical Values: (x + 4)(x - 1) = 0. Values are x = -4 and x = 1.
3. Identify Region: We want < 0 (the "valley" below the axis).
Final Answer: -4 < x < 1
Solve -x² + 4x - 7 < 0
1. Simplify: Multiply by -1 (and flip the sign) to get x² - 4x + 7 > 0.
2. Check Roots: Discriminant is 16 - 28 = -12. No real roots.
3. Sketch: The original parabola is inverted (n-shaped) and entirely below the x-axis.
Final Answer: All real numbers (x ∈ ℝ)
Solve 4x² - 16 ≥ 0
1. Find Critical Values: 4(x² - 4) = 0, so (x - 2)(x + 2) = 0. Values are x = 2 and x = -2.
2. Identify Region: We want ≥ 0 (the "tails" above the axis).
Final Answer: x ≤ -2 or x ≥ 2
Solve x² - x - 6 ≤ 0
1. Find Critical Values: (x - 3)(x + 2) = 0. Values are x = 3 and x = -2.
2. Identify Region: We want ≤ 0 (the "valley" between the roots).
Final Answer: -2 ≤ x ≤ 3
Solve (x² + 1)(x - 5) > 0
1. Analyze Factors: (x² + 1) is always positive for any real x.
2. Simplify: Since the first factor is always positive, the sign depends entirely on (x - 5).
3. Solve: x - 5 > 0.
Final Answer: x > 5
Solve x² - 10x + 25 ≤ 0
1. Find Critical Values: (x - 5)² = 0. The only value is x = 5.
2. Identify Region: A squared number is only ≤ 0 when it equals exactly zero.
Final Answer: x = 5
Solve (x² + 2x + 5)(x² - 9) < 0
1. Check first factor: x² + 2x + 5 has no real roots (discriminant = 4 - 20 = -16) and is always positive.
2. Analyse: The sign depends on (x² - 9).
3. Solve: x² - 9 < 0, which factors to (x - 3)(x + 3) < 0.
Final Answer: -3 < x < 3
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