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Form a quadratic equation in x to represent the information and find the value of x.

Mathematics

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3 parts
a

x2−5x−300=0x^2 - 5x - 300 = 0

b
20
c
15

Sh. 80\text{Sh. } 80

12x2+6x−14 km2\frac{1}{2}x^2 + 6x - 14 \text{ km}^2

Step-by-step solution

15 steps

Here are the solutions to the problems.

:

Part a· 6 steps

Form a Quadratic equation in xx to represent the information.

  1. Define variables and set up the initial equation.

    Let xx be the original number of women.
    The total amount to be contributed is Sh. 1200.
    The original contribution per woman would be 1200x\frac{1200}{x}.
    5 women were unable to contribute, so the number of contributing women became x−5x-5.
    The new contribution per woman is 1200x−5\frac{1200}{x-5}.
    The problem states that the remaining women contributed Sh. 20 more each.

    1200x−5=1200x+20\frac{1200}{x-5} = \frac{1200}{x} + 20
  2. Rearrange the equation to isolate the constant term.

    1200x−5−1200x=20\frac{1200}{x-5} - \frac{1200}{x} = 20
  3. Combine the fractions on the left side using a common denominator x(x−5)x(x-5).

    1200x−1200(x−5)x(x−5)=20\frac{1200x - 1200(x-5)}{x(x-5)} = 20 1200x−1200x+6000x2−5x=20\frac{1200x - 1200x + 6000}{x^2 - 5x} = 20 6000x2−5x=20\frac{6000}{x^2 - 5x} = 20
  4. Multiply both sides by (x2−5x)(x^2 - 5x).

    6000=20(x2−5x)6000 = 20(x^2 - 5x)
  5. Divide both sides by 20.

    600020=x2−5x\frac{6000}{20} = x^2 - 5x 300=x2−5x300 = x^2 - 5x
  6. Rearrange into the standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0.

    x2−5x−300=0x^2 - 5x - 300 = 0

    The quadratic equation is x2−5x−300=0\boxed{x^2 - 5x - 300 = 0}.

Part b20

Find the value of xx.

  1. Solve the quadratic equation x2−5x−300=0x^2 - 5x - 300 = 0 by factoring.

    We need two numbers that multiply to -300 and add to -5. These numbers are -20 and 15.

    (x−20)(x+15)=0(x - 20)(x + 15) = 0
  2. Determine the possible values for xx.

    x−20=0  ⟹  x=20x - 20 = 0 \implies x = 20 x+15=0  ⟹  x=−15x + 15 = 0 \implies x = -15
  3. Choose the valid value for xx.

    Since xx represents the number of women, it must be a positive value.
    Therefore, x=20x = 20.
    The value of xx is 20\boxed{20}.

Part c15

Determine the number of women that contributed the money and the amount each contributed.

  1. Calculate the number of women who contributed.

    The original number of women was x=20x = 20.
    5 women were unable to contribute, so the number of contributing women is x−5x - 5.
    Number of contributing women =20−5=15= 20 - 5 = 15 women.

  2. Calculate the amount each contributing woman paid.

    The total amount contributed was Sh. 1200.
    Amount each contributed =Total amountNumber of contributing women= \frac{\text{Total amount}}{\text{Number of contributing women}}
    Amount each contributed =120015=80= \frac{1200}{15} = 80 Shillings.

    The number of women that contributed the money is 15\boxed{15} and the amount each contributed is Sh. 80\boxed{\text{Sh. } 80}.

    :

    Write an expression for the area of a triangular ranch in terms of xx.

  3. Recall the formula for the area of a triangle.

    A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}
  4. Substitute the given base and height into the formula.

    Base =(x+14)= (x+14) km
    Height =(x−2)= (x-2) km

    A=12(x+14)(x−2)A = \frac{1}{2}(x+14)(x-2)
  5. Expand the product of the binomials.

    (x+14)(x−2)=x(x−2)+14(x−2)(x+14)(x-2) = x(x-2) + 14(x-2) =x2−2x+14x−28= x^2 - 2x + 14x - 28 =x2+12x−28= x^2 + 12x - 28
  6. Substitute the expanded expression back into the area formula and simplify.

    A=12(x2+12x−28)A = \frac{1}{2}(x^2 + 12x - 28) A=12x2+122x−282A = \frac{1}{2}x^2 + \frac{12}{2}x - \frac{28}{2} A=12x2+6x−14A = \frac{1}{2}x^2 + 6x - 14

    The expression for the area is 12x2+6x−14 km2\boxed{\frac{1}{2}x^2 + 6x - 14 \text{ km}^2}.

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