Let's solve this quadratic equation step-by-step by completing the square.
Given equation:
[tex]\[ 9x^2 + 49x = 22 - 5x \][/tex]
First, we need to move all terms to one side to set the equation to zero:
[tex]\[ 9x^2 + 49x + 5x - 22 = 0 \][/tex]
[tex]\[ 9x^2 + 54x - 22 = 0 \][/tex]
Next, isolate the quadratic and linear term on one side:
[tex]\[ 9x^2 + 54x = 22 \][/tex]
To complete the square, we need to factor out the coefficient of [tex]\(x^2\)[/tex] from the quadratic and linear terms:
[tex]\[ 9(x^2 + 6x) = 22 \][/tex]
Now, find the constant term that completes the square inside the parentheses. Take half of the coefficient of [tex]\(x\)[/tex] (which is 6), square it, and add it inside the parentheses. Half of 6 is 3, and [tex]\(3^2 = 9\)[/tex]:
[tex]\[ 9(x^2 + 6x + 9 - 9) = 22 \][/tex]
Rewrite the expression with the square completed:
[tex]\[ 9((x^2 + 6x + 9) - 9) = 22 \][/tex]
[tex]\[ 9((x + 3)^2 - 9) = 22 \][/tex]
Simplify the equation:
[tex]\[ 9(x + 3)^2 - 81 = 22 \][/tex]
[tex]\[ 9(x + 3)^2 = 22 + 81 \][/tex]
[tex]\[ 9(x + 3)^2 = 103 \][/tex]
So, the correct next step to solve the equation by completing the square is:
[tex]\[ 9(x^2 + 6x + 9) = 103 \][/tex]
Thus, the correct choice is:
[tex]\[ 9(x^2 + 6x + 9) = 103 \][/tex]
