Algebra (Secondary) for Competitive Exam Success

Algebra (Secondary)

3D Geometry Arithmetic and Geometric Progressions Arithmetic and Geometric Progressions - Applications Arithmetic and Geometric Progressions - Case Studies Arithmetic and Geometric Progressions - Problem Set Binomial Theorem Binomial Theorem (Intro) Binomial Theorem (Intro) - Applications Binomial Theorem (Intro) - Case Studies Binomial Theorem (Intro) - Problem Set Complex Numbers and Quadratic Equations Coordinate Geometry - Circles Coordinate Geometry - Straight Lines Differential Equations Factorization Techniques Factorization Techniques - Applications Factorization Techniques - Case Studies Factorization Techniques - Problem Set Indices and Surds Indices and Surds - Applications Indices and Surds - Case Studies Indices and Surds - Problem Set Inequalities and Their Applications Inequalities and Their Applications - Applications Inequalities and Their Applications - Case Studies Inequalities and Their Applications - Problem Set Integral Calculus Limit, Continuity and Differentiability Linear Equations in One Variable Linear Equations in One Variable - Applications Linear Equations in One Variable - Case Studies Linear Equations in One Variable - Problem Set Linear Programming Basics Linear Programming Basics - Applications Linear Programming Basics - Case Studies Linear Programming Basics - Problem Set Mathematical Induction Matrices and Determinants Pair of Linear Equations - Word Problems Pair of Linear Equations - Word Problems - Applications Pair of Linear Equations - Word Problems - Case Studies Pair of Linear Equations - Word Problems - Problem Set Permutations and Combinations Polynomials - Introduction Polynomials - Introduction - Applications Polynomials - Introduction - Case Studies Polynomials - Introduction - Problem Set Polynomials - Roots and Factor Theorem Polynomials - Roots and Factor Theorem - Applications Polynomials - Roots and Factor Theorem - Case Studies Polynomials - Roots and Factor Theorem - Problem Set Quadratic Equations Quadratic Equations - Applications Quadratic Equations - Case Studies Quadratic Equations - Problem Set Quadratic Formula Applications Quadratic Formula Applications - Applications Quadratic Formula Applications - Case Studies Quadratic Formula Applications - Problem Set Sequences and Series Sets, Relations and Functions Simultaneous Linear Equations Simultaneous Linear Equations - Applications Simultaneous Linear Equations - Case Studies Simultaneous Linear Equations - Problem Set Trigonometry - Advanced Problems Vector Algebra

Q. Factor the polynomial: 3x^2 + 12x

A. 3x(x + 4)
B. 3(x^2 + 4)
C. x(3x + 12)
D. 3x^2 + 4x

Solution

Factor out the common term 3x: 3x(x + 4).

Correct Answer: A — 3x(x + 4)

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Q. Factor the quadratic expression x^2 + 7x + 10.

A. (x + 5)(x + 2)
B. (x - 5)(x - 2)
C. (x + 10)(x - 1)
D. (x - 7)(x - 10)

Solution

We need two numbers that multiply to 10 and add to 7. The numbers 5 and 2 work. Thus, the factored form is (x + 5)(x + 2).

Correct Answer: A — (x + 5)(x + 2)

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Q. Factor the quadratic expression x^2 - 5x + 6.

A. (x - 2)(x - 3)
B. (x + 2)(x + 3)
C. (x - 1)(x - 6)
D. (x + 1)(x + 6)

Solution

To factor x^2 - 5x + 6, we need two numbers that multiply to 6 and add to -5. The numbers -2 and -3 work. Thus, the factorization is (x - 2)(x - 3).

Correct Answer: A — (x - 2)(x - 3)

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Q. Find the roots of the equation 3x^2 + 6x + 3 = 0.

A. x = -1
B. x = -3
C. x = 1
D. x = 3

Solution

This can be simplified to x^2 + 2x + 1 = 0, which factors to (x + 1)(x + 1) = 0. Thus, the root is x = -1.

Correct Answer: A — x = -1

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Q. Find the roots of the equation 3x^2 - 12 = 0.

A. x = 2, -2
B. x = 4, -4
C. x = 2, 4
D. x = -4, 2

Solution

Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.

Correct Answer: A — x = 2, -2

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Q. Find the roots of the equation 4x^2 - 12x + 9 = 0.

A. x = 1.5
B. x = 3
C. x = 0
D. x = -3

Solution

Factoring gives (2x - 3)(2x - 3) = 0. Thus, x = 3.

Correct Answer: B — x = 3

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Q. Find the roots of the equation x^2 + 2x - 8 = 0.

A. x = 2, -4
B. x = -2, 4
C. x = 4, -2
D. x = -4, 2

Solution

Factoring gives (x + 4)(x - 2) = 0. Thus, the roots are x = 4 and x = -2.

Correct Answer: C — x = 4, -2

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Q. Find the roots of the equation x^2 - 8x + 16 = 0.

A. x = 4
B. x = -4
C. x = 8
D. x = 0

Solution

This is a perfect square: (x - 4)² = 0. Thus, x - 4 = 0, so x = 4.

Correct Answer: A — x = 4

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Q. Find the roots of the polynomial equation x^2 + 5x + 6 = 0.

A. x = -2, -3
B. x = 2, 3
C. x = -1, -6
D. x = 1, -6

Solution

Factoring gives (x + 2)(x + 3) = 0. Thus, the roots are x = -2 and x = -3.

Correct Answer: A — x = -2, -3

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Q. Find the roots of the quadratic equation 3x^2 + 6x + 3 = 0.

A. x = -1
B. x = -3
C. x = 1
D. x = 3

Solution

Dividing the equation by 3 gives x^2 + 2x + 1 = 0, which factors to (x + 1)(x + 1) = 0. Thus, x = -1.

Correct Answer: A — x = -1

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Q. Find the roots of the quadratic equation 3x^2 - 12x = 0.

A. x = 0, 4
B. x = 3, 4
C. x = 0, 3
D. x = 1, 2

Solution

Factoring gives 3x(x - 4) = 0. Thus, x = 0 and x = 4.

Correct Answer: A — x = 0, 4

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Q. Find the roots of the quadratic equation 4x^2 - 12x + 9 = 0.

A. x = 1.5
B. x = 3
C. x = 0
D. x = -3

Solution

This can be factored as (2x - 3)(2x - 3) = 0. Thus, x = 3.

Correct Answer: B — x = 3

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Q. Find the solution for x in the equation 2(x + 3) = 16.

A. x = 4
B. x = 5
C. x = 6
D. x = 7

Solution

Step 1: Divide both sides by 2: x + 3 = 8. Step 2: Subtract 3 from both sides: x = 5.

Correct Answer: B — x = 5

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Q. Find the solution set for the inequality: 3x + 2 > 5.

A. x > 1
B. x < 1
C. x ≥ 1
D. x ≤ 1

Solution

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

Correct Answer: A — x > 1

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Q. Find the solution set for the inequality: 4x - 7 ≤ 9.

A. x ≤ 4
B. x ≥ 4
C. x < 4
D. x > 4

Solution

Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.

Correct Answer: A — x ≤ 4

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Q. Find the solution set for the inequality: x^2 + 2x - 8 > 0.

A. (-∞, -4) ∪ (2, ∞)
B. (-4, 2)
C. (-2, 4)
D. (-∞, 2) ∪ (4, ∞)

Solution

Step 1: Factor the quadratic: (x - 2)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (2, ∞).

Correct Answer: A — (-∞, -4) ∪ (2, ∞)

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Q. Find the solution set for the inequality: x^2 + 3x - 4 > 0.

A. (-∞, -4) ∪ (1, ∞)
B. (-4, 1)
C. (-∞, 1) ∪ (4, ∞)
D. (-4, ∞)

Solution

Step 1: Factor the quadratic: (x - 1)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (1, ∞).

Correct Answer: A — (-∞, -4) ∪ (1, ∞)

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Q. Find the solution set for the inequality: x^2 - 4 > 0.

A. (-∞, -2) ∪ (2, ∞)
B. (-2, 2)
C. (2, -2)
D. (-2, ∞)

Solution

Step 1: Factor the inequality: (x - 2)(x + 2) > 0. Step 2: The solution set is (-∞, -2) ∪ (2, ∞).

Correct Answer: A — (-∞, -2) ∪ (2, ∞)

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Q. Find the solution set for the inequality: x^2 - 6x + 8 > 0.

A. x < 2 or x > 4
B. 2 < x < 4
C. x > 2
D. x < 4

Solution

Step 1: Factor: (x - 2)(x - 4) > 0. Step 2: Critical points are x = 2 and x = 4. Step 3: Test intervals: valid for x < 2 or x > 4.

Correct Answer: A — x < 2 or x > 4

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Q. Find the solution to the inequality: -3x + 6 > 0.

A. x < 2
B. x > 2
C. x ≤ 2
D. x ≥ 2

Solution

Step 1: Subtract 6 from both sides: -3x > -6. Step 2: Divide by -3 (reverse the inequality): x < 2.

Correct Answer: B — x > 2

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Q. Find the solution to the inequality: 4x + 1 ≥ 2x + 5.

A. x ≥ 2
B. x ≤ 2
C. x ≥ 4
D. x ≤ 4

Solution

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 5. Step 2: Subtract 1: 2x ≥ 4. Step 3: Divide by 2: x ≥ 2.

Correct Answer: A — x ≥ 2

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Q. Find the solution to the inequality: 4x - 1 ≤ 3x + 2.

A. x ≤ 3
B. x ≤ 1
C. x ≥ 1
D. x ≥ 3

Solution

Step 1: Subtract 3x from both sides: x - 1 ≤ 2. Step 2: Add 1 to both sides: x ≤ 3.

Correct Answer: B — x ≤ 1

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Q. Find the solution to the inequality: 4x - 7 ≤ 9.

A. x ≤ 4
B. x ≥ 4
C. x ≤ 2
D. x ≥ 2

Solution

Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.

Correct Answer: A — x ≤ 4

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Q. Find the solution to the inequality: 4x - 7 ≥ 5.

A. x < 3
B. x > 3
C. x ≤ 3
D. x ≥ 3

Solution

Step 1: Add 7 to both sides: 4x ≥ 12. Step 2: Divide by 4: x ≥ 3.

Correct Answer: D — x ≥ 3

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Q. Find the solution to the inequality: 5 - 2x > 3.

A. x < 1
B. x > 1
C. x < -1
D. x > -1

Solution

Step 1: Subtract 5 from both sides: -2x > -2. Step 2: Divide by -2 (reverse inequality): x < 1.

Correct Answer: A — x < 1

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Q. Find the solution to the inequality: x^2 + 4x < 5.

A. (-5, 1)
B. (1, -5)
C. (1, 5)
D. (-5, 5)

Solution

Step 1: Rearrange: x^2 + 4x - 5 < 0. Step 2: Factor: (x + 5)(x - 1) < 0. Step 3: Test intervals: solution is (-5, 1).

Correct Answer: A — (-5, 1)

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Q. Find the solution to the inequality: x^2 - 9 > 0.

A. (-∞, -3) ∪ (3, ∞)
B. (-3, 3)
C. (-3, ∞)
D. (-∞, 3)

Solution

Step 1: Factor the inequality: (x - 3)(x + 3) > 0. Step 2: The critical points are x = -3 and x = 3. Step 3: Test intervals: The solution set is (-∞, -3) ∪ (3, ∞).

Correct Answer: A — (-∞, -3) ∪ (3, ∞)

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Q. Find the value of k for which the equation x^2 + kx + 9 = 0 has one real solution.

A. -18
B. -9
C. 0
D. 9

Solution

For the equation to have one real solution, the discriminant must be zero: k^2 - 4*1*9 = 0. Thus, k^2 = 36, giving k = ±6. The correct answer is -9.

Correct Answer: B — -9

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Q. Find the value of x in the equation 3x^2 + 12x + 12 = 0.

A. -2
B. -4
C. 0
D. -6

Solution

Dividing the entire equation by 3 gives x^2 + 4x + 4 = 0. Factoring gives (x + 2)(x + 2) = 0, so x = -2.

Correct Answer: B — -4

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Q. Find the value of x in the equation x^2 - 9 = 0.

A. 3
B. -3
C. 0
D. 9

Solution

Factoring gives (x - 3)(x + 3) = 0. Thus, the solutions are x = 3 and x = -3.

Correct Answer: A — 3

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Algebra (Secondary)

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