Answer :
Sure, let's express the quadratic expression [tex]\(x^2 + 12x\)[/tex] in the form [tex]\((x + a)^2 + b\)[/tex].
### Step-by-Step Solution:
1. Start with the given expression:
[tex]\[ x^2 + 12x \][/tex]
2. Identify the coefficient of [tex]\(x\)[/tex]:
The coefficient of [tex]\(x\)[/tex] is 12.
3. Divide the coefficient of [tex]\(x\)[/tex] by 2:
[tex]\[ a = \frac{12}{2} = 6 \][/tex]
Thus, [tex]\(a = 6\)[/tex].
4. Square the result from step 3:
[tex]\[ (6)^2 = 36 \][/tex]
5. Add and subtract this square within the expression:
[tex]\[ x^2 + 12x = x^2 + 12x + 36 - 36 \][/tex]
6. Rewrite the expression as a perfect square and a constant term:
[tex]\[ x^2 + 12x + 36 - 36 = (x + 6)^2 - 36 \][/tex]
Thus, we have expressed [tex]\(x^2 + 12x\)[/tex] in the form [tex]\((x + a)^2 + b\)[/tex], where:
[tex]\[ (x + 6)^2 + (-36) \][/tex]
### Conclusion:
From the above steps, we can see that the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = 6 \quad \text{and} \quad b = -36 \][/tex]
### Step-by-Step Solution:
1. Start with the given expression:
[tex]\[ x^2 + 12x \][/tex]
2. Identify the coefficient of [tex]\(x\)[/tex]:
The coefficient of [tex]\(x\)[/tex] is 12.
3. Divide the coefficient of [tex]\(x\)[/tex] by 2:
[tex]\[ a = \frac{12}{2} = 6 \][/tex]
Thus, [tex]\(a = 6\)[/tex].
4. Square the result from step 3:
[tex]\[ (6)^2 = 36 \][/tex]
5. Add and subtract this square within the expression:
[tex]\[ x^2 + 12x = x^2 + 12x + 36 - 36 \][/tex]
6. Rewrite the expression as a perfect square and a constant term:
[tex]\[ x^2 + 12x + 36 - 36 = (x + 6)^2 - 36 \][/tex]
Thus, we have expressed [tex]\(x^2 + 12x\)[/tex] in the form [tex]\((x + a)^2 + b\)[/tex], where:
[tex]\[ (x + 6)^2 + (-36) \][/tex]
### Conclusion:
From the above steps, we can see that the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = 6 \quad \text{and} \quad b = -36 \][/tex]
