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Maths expressions and equations (Secondary school)?
I have just finished a 70 page Maths booklet (that was fun...) but there are some questions that I just could not answer. All help on ANY of the questions would be great and I would prefer working out so I can understand.
1 Answer
- POMPOMLv 510 years agoFavorite Answer
Q1
8x - x² = 16 + 8x - x² - 16 = 16 - (x² - 8x + 16) = 16 - (x² - 2.x.4 + 4²) = 16 - (x - 4)²
So, p = 16, q = 4
The above expression is maximum when the (x - 4)² part vanishes.
So, the maximum value of the given expression occurs when x = 4 & the maximum value is 16.
Q2.
x² + y² = 25 ........... (1)
y = 2x - 2 ...............(2)
Put y = 2x - 2 in (1)
x² + (2x - 2)² = 25
⇒ x² + 4x² - 8x + 4 = 25
⇒ 5x² - 8x - 21 = 0
⇒ 5x² - 15x + 7x - 21 = 0
⇒ 5x(x - 3) + 7(x - 3) = 0
⇒ (x - 3)(5x + 7) = 0
⇒ x = 3, -7/5
When x = 3, y = 2x - 2 = 4
When x = -7/5, y = 2x - 2 = -24/5
So, solutions are (x = 3, y = 4) & (x = -7/5, y = -24/5)
Q3.
7/(x+2) + 1/(x-1) = 4
⇒ [7(x-1) + (x+2)]/[(x+2)(x-1)] = 4
⇒ [7x - 7 + x + 2)]/[(x+2)(x-1)] = 4
⇒ 8x - 5 = 4(x² + x -2)
⇒ 4(x² + x -2) - 8x + 5 = 0
⇒ 4x² + 4x - 8 - 8x + 5 = 0
⇒ 4x² - 4x - 3 = 0
⇒ 4x² - 6x + 2x - 3 = 0
⇒ 2x(2x - 3) + (2x - 3) = 0
⇒ (2x - 3)(2x + 1) = 0
⇒ x = 3/2, -1/2
