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f(x)=-2x^(3)+ax^(2)+2 You are told that f(2)=14, can you work out what "a" is?

f(x)=−2x3+ax2+2 f(x)=-2 x^{3}+a x^{2}+2 \newlineYou are told that f(2)=14 f(2)=14 , can you work out what

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Full solution

Q. f(x)=−2x3+ax2+2 f(x)=-2 x^{3}+a x^{2}+2 \newlineYou are told that f(2)=14 f(2)=14 , can you work out what
  1. Given Function and Condition: We are given the function f(x)=−2x3+ax2+2f(x) = -2x^3 + ax^2 + 2 and the condition f(2)=14f(2) = 14. To find the value of aa, we need to substitute x=2x = 2 into the function and set it equal to 1414.
  2. Substitute x=2x = 2: Substitute x=2x = 2 into the function: f(2)=−2(2)3+a(2)2+2f(2) = -2(2)^3 + a(2)^2 + 2.
  3. Calculate f(2)f(2): Calculate the value of f(2)f(2) with the given xx:f(2)=−2(8)+a(4)+2f(2) = -2(8) + a(4) + 2.
  4. Simplify Expression: Simplify the expression: f(2)=−16+4a+2f(2) = -16 + 4a + 2.
  5. Set Up Equation: Since we know that f(2)=14f(2) = 14, we can set up the equation: −16+4a+2=14-16 + 4a + 2 = 14.
  6. Solve for a: Solve for ＂a＂: 4a=14+16−24a = 14 + 16 - 2.
  7. Isolate aa: Simplify the right side of the equation: 4a=284a = 28.
  8. Calculate aa: Divide both sides by 44 to isolate aa:a=284.a = \frac{28}{4}.
  9. Calculate aa: Divide both sides by 44 to isolate ＂a＂\text{＂a＂}:a=284.a = \frac{28}{4}.Calculate the value of ＂a＂\text{＂a＂}:a=7.a = 7.

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