What is the first step when rewriting the expression y = 6x^2 + 18x + 14 in the form y = a(x – h)^2 + k?

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The first step in rewriting the expression y = 6x2 + 18x + 14 in the form y = a(x – h)2 + k is to identify the value of a, which is the coefficient of x2. In this case, a = 6.

Next, since we want to transform the quadratic expression into vertex form, we will complete the square for the quadratic part of the expression, which involves the terms with x: 6x2 + 18x.

To complete the square, we can factor out the coefficient of x2 (which is 6) from the first two terms:

y = 6(x2 + 3x) + 14

Now, we will focus on the expression within the parentheses: x2 + 3x. To complete the square, we need to take half of the coefficient of x (which is 3), square it (which gives us (3/2)2 = 9/4), and then add and subtract this value inside the parentheses.

This leads to:

y = 6(x2 + 3x + (9/4) - (9/4)) + 14

We can now rearrange this expression:

y = 6((x + 3/2)2 - (9/4)) + 14

By distributing the 6, we have:

y = 6(x + 3/2)2 - (54/4) + 14

Finally, we simplify the constant terms:

y = 6(x + 3/2)2 - (54/4) + (56/4)

This results in:

y = 6(x + 3/2)2 + (2/4) 
p = 6(x + 3/2)2 + (1/2)

Now, we have rewritten the original expression in the form y = a(x – h)2 + k where:

  • a = 6
  • h = -3/2 (since we have x + 3/2, we take -3/2)
  • k = 1/2

In summary, the first step is factoring out the coefficient of x2, followed by completing the square on the simplified expression.

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