Question

Rewrite the function by completing the square. f(x)=x2−9x+14f(x)= x^{2} -9 x +14 f(x)=x 2 −9x+14 f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 9, x, plus, 14 f(x)=f(x)= f(x)= f, left parenthesis, x, right parenthesis, equals (x+(x+ (x+ left parenthesis, x, plus )2+)^2+ ) 2 + right parenthesis, squared, plus

Answer 1

x² - 9x + 14 by completing the square is (x - 9/2)² - 25/4

Explanation:

Given x² - 9x + 14

To rewrite by completing the square, we need to write this in the form (a + b)² or (a - b)² without changing the value of the expression.

(a + b)² = a² + 2ab + b² (equation 1)

(a - b)² = a² - 2ab + b² (equation 2)

x² - 9x + 14 = x² - 2(x)(9/2) + (9/2)² - 25/4 (equation 3)

Comparing (equation 3) with (equation 1) and (equation 2), we can see that it takes the form of (equation 1), though, surplus of 25/4, where a = x, and b = 9/2.

So

x² - 2(x)(9/2) + (9/2)² = x² - 2(x)(9/2) + 81/4 = (x - 9/2)²

Which means

x² - 9x + 14

= x² - 2(x)(9/2) + 81/4 - 25/4

= x² - 2(x)(9/2) + (9/2)² - 25/4

= (x - 9/2)² - 25/4

And the square is completed.