To solve the given system of equations:
[tex]\[
\begin{array}{l}
y = x^2 + 5x + 3 \\
y = \sqrt{2x + 5}
\end{array}
\][/tex]
we need to find the approximate values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations.
Let's break down the solution step-by-step:
1. Set Up the Equations:
The equations given in the problem are:
[tex]\[
y = x^2 + 5x + 3 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
y = \sqrt{2x + 5} \quad \text{(Equation 2)}
\][/tex]
2. System of Equations:
We seek the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that simultaneously satisfy both equations.
3. Solve the Equations:
Through iterative methods or numerical root-finding techniques, an approximate solution is determined.
4. Approximate Solution:
Upon finding the intersection point, the values obtained are:
[tex]\[
x \approx -0.17481825533866552
\][/tex]
[tex]\[
y \approx 2.1564701457063276
\][/tex]
5. Compare with Given Options:
The possible choices are:
- A. [tex]\((2.1, 0.2)\)[/tex]
- B. [tex]\((-0.2, 2.1)\)[/tex]
- C. [tex]\((-2.1, 0.2)\)[/tex]
- D. [tex]\((0.2, 2.1)\)[/tex]
6. Determine the Closest Option:
We need to identify which of the given options is closest to the computed solution [tex]\((x, y) \approx (-0.1748, 2.1565)\)[/tex]. Option B [tex]\((-0.2, 2.1)\)[/tex] is the closest because:
- The [tex]\(x\)[/tex]-value [tex]\(-0.2\)[/tex] is very close to [tex]\(-0.1748\)[/tex].
- The [tex]\(y\)[/tex]-value [tex]\(2.1\)[/tex] is very close to [tex]\(2.1565\)[/tex].
Therefore, the solution that is closest to the computed intersection point is:
B. [tex]\((-0.2, 2.1)\)[/tex]
