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Quadratic Equations (Standard Form)

Introduction

Quadratic equations are fundamental in algebra and take the standard form ax² + bx + c = 0 where a ≠ 0. They appear in many aptitude and real-world problems-areas, projectile motion, optimization, and more.

Mastering this pattern helps you solve equations quickly using factorization, completing the square, or the quadratic formula.

Pattern: Quadratic Equations (Standard Form)

Pattern

The key concept: write the equation as ax² + bx + c = 0, compute the discriminant D = b² - 4ac, and use factorization or the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Use the discriminant to decide the nature of roots:

  • If D > 0 → two distinct real roots.
  • If D = 0 → one real repeated root.
  • If D < 0 → two complex conjugate roots.

Step-by-Step Example

Question

Solve the quadratic equation: x² - 5x + 6 = 0.

Solution

  1. Step 1: Identify coefficients

    a = 1, b = -5, c = 6.

  2. Step 2: Try factorization first

    Find two numbers whose product = a·c = 6 and whose sum = b = -5. These are -2 and -3.

  3. Step 3: Split the middle term

    x² - 5x + 6 = x² - 2x - 3x + 6.

  4. Step 4: Group and factor

    Group: (x² - 2x) - (3x - 6) = x(x - 2) - 3(x - 2) = (x - 2)(x - 3).

  5. Step 5: Use zero-product property

    (x - 2)(x - 3) = 0 ⇒ x - 2 = 0 or x - 3 = 0 ⇒ x = 2 or x = 3.

  6. Final Answer:

    x = 2, 3

  7. Quick Check:

    Substitute x = 2: 4 - 10 + 6 = 0 ✅. Substitute x = 3: 9 - 15 + 6 = 0 ✅.

Quick Variations

1. Coefficients may be fractions or negatives - multiply through by LCM if needed.

2. When factorization is hard, use the quadratic formula.

3. Completing the square is useful for deriving vertex form or solving when formula is unwieldy.

4. Word problems (areas, ages, motion) often reduce to quadratic equations.

Trick to Always Use

  • Step 1 → Check if the quadratic is easily factorable (product = a·c, sum = b).
  • Step 2 → If not factorable, compute discriminant D = b² - 4ac to decide next method.
  • Step 3 → Use the quadratic formula and simplify √D carefully; always rationalize/simplify radicals.
  • Step 4 → Quick-check each root by substitution into the original equation.

Summary

Summary

Key takeaways for Quadratic Equations (Standard Form):

  • Standardize to ax² + bx + c = 0 before solving.
  • Try factorization first for speed; if it fails, use the quadratic formula.
  • Use discriminant D = b² - 4ac to determine nature of roots.
  • Always verify roots by substitution to avoid sign mistakes.

Practice

(1/5)
1. Solve: x² - 7x + 10 = 0
easy
A. 2 and 5
B. 3 and 4
C. 1 and 10
D. 5 and 10

Solution

  1. Step 1: Identify coefficients

    a = 1, b = -7, c = 10.

  2. Step 2: Factorise

    Find two numbers whose product = 10 and sum = -7 → -2 and -5.

  3. Step 3: Write factors

    (x - 2)(x - 5) = 0.

  4. Step 4: Solve for x

    x = 2 or x = 5.

  5. Final Answer:

    x = 2, 5 → Option A.

  6. Quick Check:

    Substitute x = 2 → 4 - 14 + 10 = 0 ✅

Hint: Find two numbers whose product is c and sum is b.
Common Mistakes: Choosing wrong factor pairs or sign errors.
2. Solve: x² - 9 = 0
easy
A. 9 and -9
B. 3 and -3
C. 1 and -1
D. 0 and 9

Solution

  1. Step 1: Recognise difference of squares

    x² - 9 = (x - 3)(x + 3).

  2. Step 2: Apply zero product rule

    (x - 3)(x + 3) = 0 ⇒ x = 3 or x = -3.

  3. Final Answer:

    x = 3, -3 → Option B.

  4. Quick Check:

    Substitute x = 3 → 9 - 9 = 0 ✅

Hint: Use a² - b² = (a - b)(a + b).
Common Mistakes: Forgetting the negative root or sign errors.
3. Solve: 2x² - 8x = 0
easy
A. 0 and 2
B. 2 and 4
C. 0 and 4
D. 4 and 8

Solution

  1. Step 1: Factor out common term

    2x² - 8x = 2x(x - 4).

  2. Step 2: Set each factor = 0

    2x = 0 ⇒ x = 0; x - 4 = 0 ⇒ x = 4.

  3. Final Answer:

    x = 0, 4 → Option C.

  4. Quick Check:

    Substitute x = 4 → 2(16) - 8(4) = 32 - 32 = 0 ✅

Hint: Take out the GCF (greatest common factor) before solving.
Common Mistakes: Missing the root x = 0 after factoring out x.
4. Solve: x² - 4x + 3 = 0
medium
A. 2 and 2
B. 4 and -1
C. 0 and 3
D. 1 and 3

Solution

  1. Step 1: Identify coefficients

    a = 1, b = -4, c = 3.

  2. Step 2: Find factor pair

    Product = 3, sum = -4 → -1 and -3.

  3. Step 3: Write factors

    (x - 1)(x - 3) = 0.

  4. Step 4: Solve for x

    x = 1 or x = 3.

  5. Final Answer:

    x = 1, 3 → Option D.

  6. Quick Check:

    Substitute x = 1 → 1 - 4 + 3 = 0 ✅

Hint: Look for two numbers whose product is c and sum is b.
Common Mistakes: Sign errors when choosing factor pairs.
5. Solve: x² + 2x - 8 = 0
medium
A. 2 and -4
B. 4 and -2
C. 3 and -2
D. 1 and -8

Solution

  1. Step 1: Identify coefficients

    a = 1, b = 2, c = -8.

  2. Step 2: Factorise

    Product = -8, sum = 2 → 4 and -2.

  3. Step 3: Write factors

    (x + 4)(x - 2) = 0.

  4. Step 4: Solve for x

    x = -4 or x = 2.

  5. Final Answer:

    x = 2, -4 → Option A.

  6. Quick Check:

    Substitute x = 2 → 4 + 4 - 8 = 0 ✅

Hint: When c is negative, look for factor pairs with opposite signs to get the required sum.
Common Mistakes: Incorrect sign assignment for factor pairs.

Mock Test

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