To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the equation [tex]\(x^2 + 3x + 4 = (x + a)^2 + b\)[/tex], we'll follow these steps:
1. Expand the right-hand side of the equation [tex]\((x + a)^2 + b\)[/tex]:
[tex]\[
(x + a)^2 + b = x^2 + 2ax + a^2 + b
\][/tex]
2. Set the expanded right-hand side equal to the left-hand side of the equation:
[tex]\[
x^2 + 3x + 4 = x^2 + 2ax + a^2 + b
\][/tex]
3. Compare the coefficients of corresponding terms on both sides of the equation:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(1\)[/tex] on both sides.
- The coefficient of [tex]\(x\)[/tex] on the left-hand side is [tex]\(3\)[/tex], and on the right-hand side it is [tex]\(2a\)[/tex].
- The constant term on the left-hand side is [tex]\(4\)[/tex], and on the right-hand side it is [tex]\(a^2 + b\)[/tex].
4. Equate the coefficients to form a system of equations:
- For the [tex]\(x\)[/tex] term:
[tex]\[
2a = 3
\][/tex]
- For the constant term:
[tex]\[
a^2 + b = 4
\][/tex]
5. Solve the equation [tex]\(2a = 3\)[/tex] for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{3}{2}
\][/tex]
6. Substitute [tex]\(a = \frac{3}{2}\)[/tex] into the equation [tex]\(a^2 + b = 4\)[/tex] to find [tex]\(b\)[/tex]:
[tex]\[
\left(\frac{3}{2}\right)^2 + b = 4
\][/tex]
[tex]\[
\frac{9}{4} + b = 4
\][/tex]
[tex]\[
b = 4 - \frac{9}{4}
\][/tex]
[tex]\[
b = \frac{16}{4} - \frac{9}{4}
\][/tex]
[tex]\[
b = \frac{7}{4}
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the equation are:
[tex]\[
a = \frac{3}{2} \quad \text{and} \quad b = \frac{7}{4}
\][/tex]
