Nice problem Let y = sum of expressions in brackets divided by 2 Or y = (2x^2 -12 x +12)/ (x-3) then 2x^2 = y( x-3)+12 x-12 Numerator of first bracket is y(x-3)+3-x =( x-3)(y-1) Numerator of second bracket is y(x-3)+x-3 = (x-3)(y+1) Equation reduces to (y+1)^2+(y -1)^2 = 4 hence y = 1 , - 1 when y = 1 2 x^2 - 13 x+15 = 0 x = 5 , 3/2 when y = -1 2x^2-11 x+9 = 0 x = 1 , 9/2
Thank you for explaining. As for the final answer, -3/2 + 3 = 3/2. (In this video, -3/2 + 3 = 3. I guess it is a mistake.)
Couple of mistakes there
Nice problem
Let y = sum of expressions in brackets divided by 2
Or y = (2x^2 -12 x +12)/ (x-3)
then 2x^2 = y( x-3)+12 x-12
Numerator of first bracket is
y(x-3)+3-x =( x-3)(y-1)
Numerator of second bracket is
y(x-3)+x-3 = (x-3)(y+1)
Equation reduces to (y+1)^2+(y -1)^2 = 4
hence y = 1 , - 1
when y = 1
2 x^2 - 13 x+15 = 0
x = 5 , 3/2
when y = -1
2x^2-11 x+9 = 0
x = 1 , 9/2
[(2x² - 13x + 15)/(x - 3)]² + [(2x² - 11x + 9)/(x - 3)]² = 4 → where: x ≠ 3
[(2x² - 13x + 15)² + (2x² - 11x + 9)²]/(x - 3)² = 4
(2x² - 13x + 15)² + (2x² - 11x + 9)² = 4.(x - 3)²
4x⁴ - 26x³ + 30x² - 26x³ + 169x² - 195x + 30x² - 195x + 225 + 4x⁴ - 22x³ + 18x² - 22x³ + 121x² - 99x + 18x² - 99x + 81 = 4x² - 24x + 36
8x⁴ - 96x³ + 382x² - 564x + 270 = 0
4x⁴ - 48x³ + 191x² - 282x + 135 = 0 → the aim, if we are to continue effectively, is to eliminate terms to the 3rd power.
Let: x = z - (b/4a) → where:
b is the coefficient for x³, in our case: - 48
a is the coefficient for x⁴, in our case: 4
4x⁴ - 48x³ + 191x² - 282x + 135 = 0 → let: x = z - (- 48/16) → x = z + 3
4.(z + 3)⁴ - 48.(z + 3)³ + 191.(z + 3)² - 282.(z + 3) + 135 = 0
4.(z + 3)².(z + 3) - 48.(z + 3)².(z + 3) + 191.(z² + 6z + 9) - 282z - 846 + 135 = 0
4.(z² + 6z + 9).(z² + 6z + 9) - 48.(z² + 6z + 9).(z + 3) + 191z² + 1146z + 1719 - 282z - 711 = 0
4.(z⁴ + 6z³ + 9z² + 6z³ + 36z² + 54z + 9z² + 54z + 81) - 48.(z³ + 3z² + 6z² + 18z + 9z + 27) + 191z² + 864z + 1008 = 0
4.(z⁴ + 12z³ + 54z² + 108z + 81) - 48.(z³ + 9z² + 27z + 27) + 191z² + 864z + 1008 = 0
4z⁴ + 48z³ + 216z² + 432z + 324 - 48z³ - 432z² - 1296z - 1296 + 191z² + 864z + 1008 = 0
4z⁴ - 25z² + 36 = 0 → by completing the square: 4z⁴ - 25z² = [2z² - (25/4)]² - (25/4)²
[2z² - (25/4)]² - (25/4)² + 36 = 0
[2z² - (25/4)]² - (625/16) + (576/16) = 0
[2z² - (25/4)]² - (49/16) = 0
[2z² - (25/4)]² = 49/16
[2z² - (25/4)]² = (7/4)²
2z² - (25/4) = ± 7/4
2z² = (25/4) ± (7/4)
2z² = (25 ± 7)/4
z² = (25 ± 7)/8
First case: z² = (25 + 7)/8
z² = 32/8
z² = 4
z = ± 2 → recall: x = z + 3
x = ± 2 + 3
→ x = 1
→ x = 5
Second case: z² = (25 - 7)/8
z² = 18/8
z² = 9/4
z = ± 3/2 → recall: x = z + 3
x = 3 ± (3/2)
→ x = 9/2
→ x = 3/2