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Sagot :
To solve the given problem, we start by establishing the relationships described in the problem:
1. Cost of eggs: Sirak buys \( x \) eggs at \( (x - 8) \) cents each. Therefore, the total cost of the eggs is:
[tex]\[ \text{Cost of eggs} = x \times (x - 8) = x^2 - 8x \text{ cents} \][/tex]
2. Cost of bread rolls: Sirak buys \( (x - 2) \) bread rolls at \( (x - 3) \) cents each. Therefore, the total cost of the bread rolls is:
[tex]\[ \text{Cost of bread rolls} = (x - 2) \times (x - 3) = x^2 - 5x + 6 \text{ cents} \][/tex]
3. Total bill: The total cost in cents is:
[tex]\[ \text{Total cost} = \left( x^2 - 8x \right) + \left( x^2 - 5x + 6 \right) \][/tex]
The total bill equals $1.75, converted into cents:
[tex]\[ 1.75 \text{ dollars} = 1.75 \times 100 = 175 \text{ cents} \][/tex]
4. Equation: Adding the cost of eggs and bread rolls, we set up the following equation:
[tex]\[ \left( x^2 - 8x \right) + \left( x^2 - 5x + 6 \right) = 175 \][/tex]
Simplify and combine like terms:
[tex]\[ 2x^2 - 13x + 6 = 175 \][/tex]
Subtract 175 from both sides to set the equation to zero:
[tex]\[ 2x^2 - 13x + 6 - 175 = 0 \][/tex]
[tex]\[ 2x^2 - 13x - 169 = 0 \][/tex]
5. Solve the quadratic equation: We solve the quadratic equation:
[tex]\[ 2x^2 - 13x - 169 = 0 \][/tex]
Solving this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -13\), and \(c = -169\), we get:
[tex]\[ x = \frac{13 \pm \sqrt{(-13)^2 - 4(2)(-169)}}{2(2)} \][/tex]
[tex]\[ x = \frac{13 \pm \sqrt{169 + 1352}}{4} \][/tex]
[tex]\[ x = \frac{13 \pm \sqrt{1521}}{4} \][/tex]
[tex]\[ x = \frac{13 \pm 39}{4} \][/tex]
Calculating the possible values for \( x \):
[tex]\[ x = \frac{13 + 39}{4} = \frac{52}{4} = 13 \][/tex]
[tex]\[ x = \frac{13 - 39}{4} = \frac{-26}{4} = -6.5 \][/tex]
6. Validate the solutions: Since \( x \) must be a positive integer representing the number of eggs:
- \( x = 13 \) is a valid solution.
- \( x = -6.5 \) is not valid as it is not a positive integer.
Therefore, Sirak buys [tex]\( \boxed{13} \)[/tex] eggs.
1. Cost of eggs: Sirak buys \( x \) eggs at \( (x - 8) \) cents each. Therefore, the total cost of the eggs is:
[tex]\[ \text{Cost of eggs} = x \times (x - 8) = x^2 - 8x \text{ cents} \][/tex]
2. Cost of bread rolls: Sirak buys \( (x - 2) \) bread rolls at \( (x - 3) \) cents each. Therefore, the total cost of the bread rolls is:
[tex]\[ \text{Cost of bread rolls} = (x - 2) \times (x - 3) = x^2 - 5x + 6 \text{ cents} \][/tex]
3. Total bill: The total cost in cents is:
[tex]\[ \text{Total cost} = \left( x^2 - 8x \right) + \left( x^2 - 5x + 6 \right) \][/tex]
The total bill equals $1.75, converted into cents:
[tex]\[ 1.75 \text{ dollars} = 1.75 \times 100 = 175 \text{ cents} \][/tex]
4. Equation: Adding the cost of eggs and bread rolls, we set up the following equation:
[tex]\[ \left( x^2 - 8x \right) + \left( x^2 - 5x + 6 \right) = 175 \][/tex]
Simplify and combine like terms:
[tex]\[ 2x^2 - 13x + 6 = 175 \][/tex]
Subtract 175 from both sides to set the equation to zero:
[tex]\[ 2x^2 - 13x + 6 - 175 = 0 \][/tex]
[tex]\[ 2x^2 - 13x - 169 = 0 \][/tex]
5. Solve the quadratic equation: We solve the quadratic equation:
[tex]\[ 2x^2 - 13x - 169 = 0 \][/tex]
Solving this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -13\), and \(c = -169\), we get:
[tex]\[ x = \frac{13 \pm \sqrt{(-13)^2 - 4(2)(-169)}}{2(2)} \][/tex]
[tex]\[ x = \frac{13 \pm \sqrt{169 + 1352}}{4} \][/tex]
[tex]\[ x = \frac{13 \pm \sqrt{1521}}{4} \][/tex]
[tex]\[ x = \frac{13 \pm 39}{4} \][/tex]
Calculating the possible values for \( x \):
[tex]\[ x = \frac{13 + 39}{4} = \frac{52}{4} = 13 \][/tex]
[tex]\[ x = \frac{13 - 39}{4} = \frac{-26}{4} = -6.5 \][/tex]
6. Validate the solutions: Since \( x \) must be a positive integer representing the number of eggs:
- \( x = 13 \) is a valid solution.
- \( x = -6.5 \) is not valid as it is not a positive integer.
Therefore, Sirak buys [tex]\( \boxed{13} \)[/tex] eggs.
Answer:
13 eggs
Step-by-step explanation:
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