Answer :
To solve the quadratic equation [tex]\(0 = 9(x^2 + 6x) - 18\)[/tex] by completing the square, we can follow these steps:
1. Distribute the 9 and move the constant to the other side of the equation:
[tex]\[ 0 = 9(x^2 + 6x) - 18 \][/tex]
Add 18 to both sides:
[tex]\[ 18 = 9(x^2 + 6x) \][/tex]
2. Divide by 9 on both sides to simplify:
[tex]\[ \frac{18}{9} = x^2 + 6x \][/tex]
[tex]\[ 2 = x^2 + 6x \][/tex]
3. Complete the square on the right-hand side:
To complete the square, take half of the coefficient of [tex]\(x\)[/tex] (which is 6), square it, and add it to both sides:
[tex]\[ 2 + 9 = x^2 + 6x + 9 \][/tex]
[tex]\[ 11 = (x + 3)^2 \][/tex]
4. Solve the resulting equation by taking the square root of both sides:
[tex]\[ \sqrt{11} = x + 3 \][/tex]
Therefore,
[tex]\[ x = \sqrt{11} - 3 \][/tex]
or
[tex]\[ x = -\sqrt{11} - 3 \][/tex]
Given the provided options, we can identify the steps that are valid and correspond to the correct procedure:
- [tex]\(18=9(x^2+6x)\)[/tex]
- [tex]\(18+9 = 9(x^2 + 6x + 9)\)[/tex] (completing the square using [tex]\((6/2)^2 = 9\)[/tex])
- [tex]\(11 = (x + 3)^2\)[/tex] (reflects the square completion correctly)
Therefore, the valid steps are:
[tex]\[ \boxed{18 + 9 = 9(x^2 + 6x + 9)} \][/tex]
[tex]\[ \boxed{11 = (x + 3)^2} \][/tex]
1. Distribute the 9 and move the constant to the other side of the equation:
[tex]\[ 0 = 9(x^2 + 6x) - 18 \][/tex]
Add 18 to both sides:
[tex]\[ 18 = 9(x^2 + 6x) \][/tex]
2. Divide by 9 on both sides to simplify:
[tex]\[ \frac{18}{9} = x^2 + 6x \][/tex]
[tex]\[ 2 = x^2 + 6x \][/tex]
3. Complete the square on the right-hand side:
To complete the square, take half of the coefficient of [tex]\(x\)[/tex] (which is 6), square it, and add it to both sides:
[tex]\[ 2 + 9 = x^2 + 6x + 9 \][/tex]
[tex]\[ 11 = (x + 3)^2 \][/tex]
4. Solve the resulting equation by taking the square root of both sides:
[tex]\[ \sqrt{11} = x + 3 \][/tex]
Therefore,
[tex]\[ x = \sqrt{11} - 3 \][/tex]
or
[tex]\[ x = -\sqrt{11} - 3 \][/tex]
Given the provided options, we can identify the steps that are valid and correspond to the correct procedure:
- [tex]\(18=9(x^2+6x)\)[/tex]
- [tex]\(18+9 = 9(x^2 + 6x + 9)\)[/tex] (completing the square using [tex]\((6/2)^2 = 9\)[/tex])
- [tex]\(11 = (x + 3)^2\)[/tex] (reflects the square completion correctly)
Therefore, the valid steps are:
[tex]\[ \boxed{18 + 9 = 9(x^2 + 6x + 9)} \][/tex]
[tex]\[ \boxed{11 = (x + 3)^2} \][/tex]
