Exercise 3. 10 Math Magic ✨

Step 1: Multiply coefficient of x² with constant term
4 × (-2) = -8

Step 2: Find two numbers that multiply to -8 and add to -7
The numbers are -8 and +1 because (-8)×1 = -8 and -8+1 = -7

Step 3: Rewrite the middle term using these numbers
4x² - 8x + 1x - 2 = 0

Step 4: Factor by grouping
(4x² - 8x) + (1x - 2) = 0 → 4x(x - 2) + 1(x - 2) = 0
(4x + 1)(x - 2) = 0

Final Answer:
x = -1/4 or x = 2

Step 1: Expand both sides
3p² - 18 = p² + 5p

Step 2: Bring all terms to one side
3p² - p² - 5p - 18 = 0 → 2p² - 5p - 18 = 0

Step 3: Multiply coefficient of p² with constant term
2 × (-18) = -36

Step 4: Find two numbers that multiply to -36 and add to -5
The numbers are -9 and +4 because (-9)×4 = -36 and -9+4 = -5

Step 5: Rewrite and factor
2p² - 9p + 4p - 18 = 0 → p(2p - 9) + 2(2p - 9) = 0 → (p + 2)(2p - 9) = 0

Final Answer:
p = -2 or p = 9/2

Step 1: Expand the left side
a² - 7a = 32

Step 2: Bring all terms to one side
a² - 7a - 32 = 0

Step 3: Find two numbers that multiply to -32 and add to -7
The numbers are -√32 and +√32, but they don't add to -7, so we'll use quadratic formula

Step 4: Apply quadratic formula: a = [7 ± √(49 + 128)]/2
a = [7 ± √177]/2

Final Answer:
a = (7 + √177)/2 or a = (7 - √177)/2

Step 1: Multiply coefficient of x² with constant term
2 × 5√2 = 10√2

Step 2: Find two numbers that multiply to 10√2 and add to 7
The numbers are 5√2 and 2 because 5√2 × 2 = 10√2 and 5√2 + 2 ≈ 7 (since √2 ≈ 1.414)

Step 3: Rewrite the middle term
2x² + 5√2x + 2x + 5√2 = 0

Step 4: Factor by grouping
(2x² + 2x) + (5√2x + 5√2) = 0 → 2x(x + 1) + 5√2(x + 1) = 0 → (2x + 5√2)(x + 1) = 0

Final Answer:
x = -5√2/2 or x = -1

Step 1: Multiply all terms by 8 to eliminate fraction
x² - 8x + 16 = 0

Step 2: This is a perfect square trinomial
(x - 4)² = 0

Final Answer:
x = 4 (double root)