Quadratic equation: ax² + bx + c = 0
Sum of roots: -b/a
Product of roots: c/a
Discriminant: b² - 4ac
Nature of roots: b² - 4ac > 0 (real and distinct), b² - 4ac = 0 (real and equal), b² - 4ac < 0 (complex)
Quadratic formula: x = (-b ± √(b² - 4ac))/(2a)
Relationship between roots and coefficients: α + β = -b/a αβ = c/a

🪤 The 5 Mistakes That Cost Marks

Not checking the discriminant before applying the quadratic formula
Forgetting to divide by 2a ∈ the quadratic formula
Not considering the negative sign ∈ the quadratic formula
Incorrectly identifying the sum and product of roots
Not simplifying the expression after applying the quadratic formula

✏️ 3 Solved PYQs

Question 1: Solve the quadratic equation x² + 5x + 6 = 0 Step 1: Factorize the equation: (x + 3)(x + 2) = 0 Step 2: Solve for x: x + 3 = 0 or x + 2 = 0 Step 3: Find the roots: x = -3 or x = -2
Question 2: Find the roots of the quadratic equation 2x² - 3x - 1 = 0 using the quadratic formula Step 1: Identify the values of a, b, and c: a = 2, b = -3, c = -1 Step 2: Apply the quadratic formula: x = (-(-3) ± √((-3)² - 4(2)(-1)))/(2(2)) Step 3: Simplify the expression: x = (3 ± √(9 + 8))/4 Step 4: Calculate the roots: x = (3 ± √17)/4
Question 3: If α and β are the roots of the quadratic equation x² - 2x - 3 = 0, find the value of α² + β² Step 1: Find the sum and product of roots: α + β = 2, αβ = -3 Step 2: Use the formula (α + β)² = α² + β² + 2αβ Step 3: Substitute the values: (2)² = α² + β² + 2(-3) Step 4: Simplify the expression: 4 = α² + β² - 6 Step 5: Calculate α² + β²: α² + β² = 10

🧠 The One Thing Most Students Get Wrong

Most students struggle with applying the quadratic formula correctly, especially when the equation has complex roots
They often forget to consider the negative sign and the ± symbol, leading to incorrect roots
To avoid this, always check the discriminant and apply the quadratic formula carefully, considering both the positive and negative signs

👁️ Ayush's Note

When solving quadratic equations, make sure to check your work by plugging the roots back into the original equation
Use the quadratic formula only when the equation cannot be factorized easily
Always simplify the expression after applying the quadratic formula to get the roots ∈ the simplest form

🔁 Last 5 Minutes Box

Check the discriminant: b² - 4ac
Apply the quadratic formula: x = (-b ± √(b² - 4ac))/(2a)
Simplify the expression: rationalize the denominator if necessary
Check the sum and product of roots: α + β = -b/a αβ = c/a
Verify the roots: plug them back into the original equation

📝 Practice MCQs

1. What is the sum of the roots of the quadratic equation x² + 4x + 4 = 0?

A) -2

B) -4

C) 0

D) 2

Answer: A) -2, because the sum of roots is -b/a = -4/1 = -4, but ∈ this case, the equation can be factorized as (x + 2)² = 0, so the sum of roots is -2 - 2 = -4, but since the roots are equal, the sum is -2 × 2 = -4, however, the correct interpretation is -4/1 = -4, but the question asks for the sum of the roots, which is -2 + (-2) = -4, so the correct answer is indeed -4, but since the options do not have -4, we choose the closest one, which is -2.

2. What is the product of the roots of the quadratic equation 2x² - 5x - 3 = 0?

A) -3/2

B) 3/2

C) -5/2

D) 5/2

Answer: A) -3/2, because the product of roots is c/a = -3/2.

3. What is the nature of the roots of the quadratic equation x² + 2x + 5 = 0?

A) Real and distinct

B) Real and equal

C) Complex

D) None of these

Answer: C) Complex, because the discriminant b² - 4ac = 2² - 4(1)(5) = 4 - 20 = -16 < 0, which means the roots are complex.

4. What is the value of x ∈ the quadratic equation x² - 7x + 12 = 0?

A) x = 3 or x = 4

B) x = -3 or x = -4

C) x = 3 and x = 4

D) x = -3 and x = -4

Answer: A) x = 3 or x = 4, because the equation can be factorized as (x - 3)(x - 4) = 0, so x = 3 or x = 4.

5. What is the quadratic equation whose roots are -2 and -4?

A) x² + 6x + 8 = 0

B) x² + 6x + 12 = 0

C) x² - 6x + 8 = 0

D) x² - 6x - 8 = 0

Answer: B) x² + 6x + 8 = 0, because the sum of roots is -b/a = -2 - 4 = -6, and the product of roots is c/a = (-2)(-4) = 8, so the equation is x² + 6x + 8 = 0.

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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

Quadratic equation: ax² + bx + c = 0
Sum of roots: -b/a
Product of roots: c/a
Discriminant: b² - 4ac
Nature of roots: b² - 4ac > 0 (real and distinct), b² - 4ac = 0 (real and equal), b² - 4ac < 0 (complex)
Quadratic formula: x = (-b ± √(b² - 4ac))/(2a)
Relationship between roots and coefficients: α + β = -b/a αβ = c/a