To make the expressions perfect squares, the value of c should be the square of half the coefficient of the x term. Therefore, the values of c are 9, 49, 36, 25, 1, and 1600, respectively, for each of the given expressions.
To determine the value of c that makes each expression a perfect square, we need to find a number that when added to the quadratic expression x2 + bx, allows it to be factored into (x + d)2.
For a quadratic expression of the form x2 + bx + c, the value of c is obtained by squaring half of the coefficient b from the x term.
This process is called completing the square.
For x2 + 6x + c, c = (6/2)2 = 9
For x2 + 14x + c, c = (14/2)2 = 49
For x2 - 12x + c, c = (-12/2)2 = 36
For x2 - 10x + c, c = (-10/2)2 = 25
For x2 + 2x + c, c = (2/2)2 = 1
For x2 - 80x + c, c = (-80/2)2 = 1600