Question

If \( f(x+5)=x^{2}-5 x+7 \) find \( f(x) \) 

A. \( f(x)=x^{2}-15 x+37 \)

B. \( f(x)=x^{2}+15 x+57 \) 

C. \( f(x)=x^{2}-15 x+37 \)

D. \( f(x)=x^{2}-15 x+57 \)

Answer 1

Correct answer is:- (D) f(x) = x² - 15x + 57

Explanation:- Let, 

f(x) = ax² + bx + c... (1)  

Now, f(x + 5) = a(x + 5)² + b(x + 5) + c

x² - 5x + 7 = ax² +10ax + 25a + bx + 5b + c. 

Comparing the coefficient of x², we get a = 1, 

x² - 5x + 7 = x² + 10x + 25 + bx + 5b + c

-15x - 18 = bx + 5b + c, 

Comparing the coefficient of x, we get b = -15, 

-15x - 18 = -15x - 75 + c

c = 57. 

Putting the values of a, b and c in (1) we get, 

f(x) = x² - 15x + 57. 

Therefore, the required polynomial is x² - 15x + 57 

Comments

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Answer 2

(D) x² - 15x + 57

f(x + 5) = x² - 5x + 7

\(\therefore\) f(x) = (x - 5)² - 5(x - 5) + 7 (By taking x -5 as x)

 = x² - 10x + 25 - 5x + 25 + 7

 = x² - 15x + 57