Quadratic Equation Adventure

Solve equations like a math wizard! ✨

Remember the Quadratic Formula:

x = [ -b ± √(b² - 4ac) ] / 2a

🔍1. Completing the Square Method

📌Problem (i): 9x² - 12x + 4 = 0
1 Divide by coefficient of x²:
x² - (12/9)x + 4/9 = 0 → x² - (4/3)x + 4/9 = 0
2 Move constant term:
x² - (4/3)x = -4/9
3 Complete the square:
Take half of 4/3 → 2/3, square it → 4/9
x² - (4/3)x + 4/9 = -4/9 + 4/9
4 Simplify:
(x - 2/3)² = 0
5 Solve for x:
x - 2/3 = 0 → x = 2/3
🎉 Final Answer:
x = 2/3 (double root)
📌Problem (ii): (5x + 7)/(x - 1) = 3x + 2
1 Multiply both sides by (x - 1):
5x + 7 = (3x + 2)(x - 1)
2 Expand right side:
5x + 7 = 3x² - 3x + 2x - 2 → 5x + 7 = 3x² - x - 2
3 Bring all terms to one side:
0 = 3x² - x - 2 - 5x - 7 → 3x² - 6x - 9 = 0
4 Simplify by dividing by 3:
x² - 2x - 3 = 0
5 Complete the square:
x² - 2x = 3 → x² - 2x + 1 = 3 + 1 → (x - 1)² = 4
6 Solve for x:
x - 1 = ±2 → x = 1 ± 2
🎉 Final Answers:
x = 3 or x = -1

🧮2. Quadratic Formula Method

📌Problem (i): 2x² - 5x + 2 = 0
1 Identify coefficients:
a = 2, b = -5, c = 2
2 Calculate discriminant (D):
D = b² - 4ac = (-5)² - 4(2)(2) = 25 - 16 = 9
3 Apply quadratic formula:
x = [ -(-5) ± √9 ] / (2×2) = [ 5 ± 3 ] / 4
4 Calculate both solutions:
x₁ = (5 + 3)/4 = 8/4 = 2
x₂ = (5 - 3)/4 = 2/4 = 0.5
🎉 Final Answers:
x = 2 or x = 0.5
📌Problem (ii): √2 f² - 6f + 3√2 = 0
1 Identify coefficients:
a = √2, b = -6, c = 3√2
2 Calculate discriminant (D):
D = (-6)² - 4(√2)(3√2) = 36 - 4(3)(2) = 36 - 24 = 12
3 Apply quadratic formula:
f = [ -(-6) ± √12 ] / (2×√2) = [ 6 ± 2√3 ] / 2√2
4 Simplify:
f = (6 ± 2√3)/(2√2) = (3 ± √3)/√2
Rationalize: [(3 ± √3)√2]/2 = (3√2 ± √6)/2
🎉 Final Answers:
f = (3√2 + √6)/2 or f = (3√2 - √6)/2
📌Problem (iii): 3y² - 20y - 23 = 0
1 Identify coefficients:
a = 3, b = -20, c = -23
2 Calculate discriminant (D):
D = (-20)² - 4(3)(-23) = 400 + 276 = 676
3 Apply quadratic formula:
y = [ -(-20) ± √676 ] / (2×3) = [ 20 ± 26 ] / 6
4 Calculate both solutions:
y₁ = (20 + 26)/6 = 46/6 = 23/3 ≈ 7.666
y₂ = (20 - 26)/6 = -6/6 = -1
🎉 Final Answers:
y = 23/3 or y = -1
📌Problem (iv): 36y² - 12ay + (a² - b²) = 0
1 Identify coefficients:
a = 36, b = -12a, c = (a² - b²)
2 Calculate discriminant (D):
D = (-12a)² - 4(36)(a² - b²) = 144a² - 144(a² - b²) = 144a² - 144a² + 144b² = 144b²
3 Apply quadratic formula:
y = [ -(-12a) ± √(144b²) ] / (2×36) = [ 12a ± 12|b| ] / 72
4 Simplify:
y = [12(a ± |b|)]/72 = (a ± |b|)/6
🎉 Final Answers:
y = (a + b)/6 or y = (a - b)/6

⚽3. Real-world Application

📌Ball Rolling Problem

A ball rolls down a slope with distance equation: d = t² - 0.75t feet in t seconds. Find when d = 11.25 feet.

1 Set up equation:
t² - 0.75t = 11.25 → t² - 0.75t - 11.25 = 0
2 Identify coefficients:
a = 1, b = -0.75, c = -11.25
3 Calculate discriminant (D):
D = (-0.75)² - 4(1)(-11.25) = 0.5625 + 45 = 45.5625
4 Apply quadratic formula:
t = [ -(-0.75) ± √45.5625 ] / 2 = [ 0.75 ± 6.75 ] / 2
5 Calculate both solutions:
t₁ = (0.75 + 6.75)/2 = 7.5/2 = 3.75 seconds
t₂ = (0.75 - 6.75)/2 = -6/2 = -3 seconds
6 Reject negative time:
Time cannot be negative, so we discard t = -3
🎉 Final Answer:
The ball travels 11.25 feet after 3.75 seconds

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