Complex Numbers and Quadratic Equations Ex 5.3 Class 11 NCERT

The Quadratic Formula in the Complex Realm

Let's start with a familiar friend: the standard quadratic equation, ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. You know the powerful quadratic formula to find its roots:

x = [-b ± sqrt(b² - 4ac)] / 2a

The expression inside the square root, D = b² - 4ac, is called the discriminant. In Class 10, we learned:

• If D > 0, there are two distinct real roots.
• If D = 0, there is exactly one real root (or two equal real roots).
• If D < 0, there are no real roots.

Exercise 5.3 is all about this third case. Now that we have complex numbers, we can handle a negative discriminant. The key is understanding the imaginary unit i, which is defined as i = sqrt(-1). This allows us to take the square root of any negative number. For example, sqrt(-9) = sqrt(9 -1) = sqrt(9) sqrt(-1) = 3i.

So, when D < 0, our discriminant is a negative number. We can rewrite sqrt(D) as sqrt(-1 |D|) = sqrt(-1) sqrt(|D|) = i sqrt(|D|). Here, |D| is the positive value of D. For instance, if D = -7, then |D| = 7 and sqrt(-7) = i sqrt(7). Our quadratic formula thus transforms for complex roots:

x = [-b ± i * sqrt(-(b² - 4ac))] / 2a

Notice that the roots will be a pair of complex conjugates: (-b/2a) + i(sqrt(-D)/2a) and (-b/2a) - i(sqrt(-D)/2a). This is the core concept you'll apply to every problem in this exercise.

Step-by-Step Guide to Finding Complex Roots

1. Step 1: Identify Coefficients — Given a quadratic equation ax² + bx + c = 0, clearly identify the values of a, b, and c. Be careful with signs!
2. Step 2: Calculate the Discriminant (D) — Use the formula D = b² - 4ac to calculate the discriminant. For this exercise, you will find that D < 0.
3. Step 3: Apply the Complex Root Formula — Substitute the values of a, b, and the now-negative D into the quadratic formula. Remember that sqrt(D) can be written as i * sqrt(-D), where -D is a positive value.
4. Step 4: Simplify and State the Roots — Simplify the expression to get the two roots. The final answer should be in the standard complex form x = p ± iq. These two roots will always be complex conjugates of each other.

Worked Examples from Ex 5.3

• Problem: Solve the equation x² + 3 = 0. Step 1: Identify Coefficients Here, the equation is 1x² + 0x + 3 = 0. So, a = 1, b = 0, and c = 3. Step 2: Calculate the Discriminant (D) D = b² - 4ac = (0)² - 4(1)(3) = 0 - 12 = -12. Since D < 0, the roots are complex. Step 3: Apply the Formula x = [-b ± sqrt(D)] / 2a = [-(0) ± sqrt(-12)] / 2(1) x = ± sqrt(-12) / 2 Step 4: Simplify and State the Roots We need to simplify sqrt(-12). sqrt(-12) = sqrt(12 -1) = sqrt(12) sqrt(-1) = sqrt(4 3) i = 2√3 i. Substituting this back: x = ± (2√3 i) / 2 x = ± √3 i The solutions are x = √3 i and x = -√3 i.
• Problem: Solve the equation 2x² + x + 1 = 0. Step 1: Identify Coefficients Comparing with ax² + bx + c = 0, we have a = 2, b = 1, and c = 1. Step 2: Calculate the Discriminant (D) D = b² - 4ac = (1)² - 4(2)(1) = 1 - 8 = -7. Since D < 0, the roots are complex. Step 3: Apply the Formula x = [-b ± sqrt(D)] / 2a = [-(1) ± sqrt(-7)] / 2(2) x = [-1 ± sqrt(-7)] / 4 Step 4: Simplify and State the Roots We write sqrt(-7) as i√7. x = [-1 ± i√7] / 4 We can write this as two separate roots to make it clear: The solutions are x = -1/4 + (√7/4)i and x = -1/4 - (√7/4)i.

Exam Traps and Common Mistakes

Watch out for these common errors when solving for complex roots:

• Forgetting the 'i': A very frequent mistake is calculating sqrt(-D) as just sqrt(D). If D = -25, the square root is 5i, not just 5. Remember, a negative inside a square root always produces an i in the result.
• Sign Errors in Discriminant: Be extra careful with negative signs when calculating b² - 4ac. If b = -3, then b² = (-3)² = +9, not -9. If c is negative, the -4ac term will become positive.
• Incomplete Simplification: When you have a result like x = (-4 ± 6i) / 2, you must divide both the real part (-4) and the imaginary part (6i) by the denominator (2). The correct simplified answer is x = -2 ± 3i, not -2 ± 6i or (-4 ± 3i) / 2.

Practice Questions with Solutions

• Q: Solve the quadratic equation: x² + x + 1 = 0 A: Step 1: Identify coefficients. Here, a = 1, b = 1, c = 1. Step 2: Calculate the discriminant. D = b² - 4ac = 1² - 4(1)(1) = 1 - 4 = -3. Step 3: Apply the quadratic formula. x = [-b ± sqrt(D)] / 2a = [-1 ± sqrt(-3)] / 2(1). Step 4: Simplify. sqrt(-3) = i√3. So, x = [-1 ± i√3] / 2. Final answer: The roots are -1/2 + i√3/2 and -1/2 - i√3/2.
• Q: Solve the quadratic equation: √2x² + x + √2 = 0 A: Step 1: Identify coefficients. Here, a = √2, b = 1, c = √2. Step 2: Calculate the discriminant. D = b² - 4ac = 1² - 4(√2)(√2) = 1 - 4(2) = 1 - 8 = -7. Step 3: Apply the quadratic formula. x = [-b ± sqrt(D)] / 2a = [-1 ± sqrt(-7)] / (2√2). Step 4: Simplify. sqrt(-7) = i√7. So, x = [-1 ± i√7] / (2√2). Final answer: The roots are (-1 / 2√2) + i(√7 / 2√2) and (-1 / 2√2) - i(√7 / 2√2).
• Q: Solve the quadratic equation: x² + 3x + 5 = 0 A: Step 1: Identify coefficients. Here, a = 1, b = 3, c = 5. Step 2: Calculate the discriminant. D = b² - 4ac = 3² - 4(1)(5) = 9 - 20 = -11. Step 3: Apply the quadratic formula. x = [-b ± sqrt(D)] / 2a = [-3 ± sqrt(-11)] / 2(1). Step 4: Simplify. sqrt(-11) = i√11. So, x = [-3 ± i√11] / 2. Final answer: The roots are -3/2 + i√11/2 and -3/2 - i√11/2.
• Q: Solve the quadratic equation: -x² + x - 2 = 0 A: Step 1: It's easier to work with a positive a. Multiply the whole equation by -1: x² - x + 2 = 0. Now, a = 1, b = -1, c = 2. Step 2: Calculate the discriminant. D = b² - 4ac = (-1)² - 4(1)(2) = 1 - 8 = -7. Step 3: Apply the quadratic formula. x = [-b ± sqrt(D)] / 2a = [-(-1) ± sqrt(-7)] / 2(1) = [1 ± sqrt(-7)] / 2. Step 4: Simplify. sqrt(-7) = i√7. So, x = [1 ± i√7] / 2. Final answer: The roots are 1/2 + i√7/2 and 1/2 - i√7/2.

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