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A function p is defined as p(x)=(x-a)(x-15)(x-20) where a is a constant. Given that p(7)=15, what is the value of a ?

A function p p is defined as p(x)=(x−a)(x−15)(x−20) p(x)=(x-a)(x-15)(x-20) where a a is a constant. Given that p(7)=15 p(7)=15 , what is the value of a a ?

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Q. A function p p is defined as p(x)=(x−a)(x−15)(x−20) p(x)=(x-a)(x-15)(x-20) where a a is a constant. Given that p(7)=15 p(7)=15 , what is the value of a a ?
  1. Substitute and Set Equal: To find the value of aa, we need to substitute x=7x = 7 into the function p(x)p(x) and set it equal to 1515, as given by the condition p(7)=15p(7) = 15.\newlineCalculation: p(7)=(7−a)(7−15)(7−20)=15p(7) = (7 - a)(7 - 15)(7 - 20) = 15
  2. Simplify Expression: Simplify the expression by calculating the known values.\newlineCalculation: (7−a)(−8)(−13)=15(7 - a)(-8)(-13) = 15
  3. Multiply Constants: Multiply the constants (−8)(-8) and (−13)(-13) to simplify the equation further.\newlineCalculation: (7−a)(104)=15(7 - a)(104) = 15
  4. Divide to Isolate: Divide both sides of the equation by 104104 to isolate (7−a)(7 - a).\newlineCalculation: 7−a=151047 - a = \frac{15}{104}
  5. Simplify Right Side: Simplify the right side of the equation.\newlineCalculation: 7−a=15104≈0.1442307697 - a = \frac{15}{104} \approx 0.144230769
  6. Subtract and Solve: Subtract 77 from both sides to solve for aa.\newlineCalculation: −a=0.144230769−7-a = 0.144230769 - 7
  7. Solve for a: Solve for a by multiplying both sides by −1-1.\newlineCalculation: a=7−0.144230769a = 7 - 0.144230769
  8. Complete Calculation: Complete the calculation to find the value of aa.\newlineCalculation: a≈7−0.144230769≈6.855769231a \approx 7 - 0.144230769 \approx 6.855769231

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