Let's solve the given equation:
[tex]\[
x^2 - 5x + 6 = \sqrt{x + 3}
\][/tex]
1. Set Up the Equation:
We are given the equation:
[tex]\[
x^2 - 5x + 6 = \sqrt{x + 3}
\][/tex]
2. Graphical Approach:
To find where the left-hand side (LHS) and the right-hand side (RHS) are equal, it helps to visualize the problem. The LHS is a quadratic equation [tex]\(y = x^2 - 5x + 6\)[/tex]. The RHS is a square root function [tex]\( y = \sqrt{x + 3} \)[/tex].
3. Approximate Solutions Calculation:
From the graphical perspective, the intersections of these curves give the solutions.
By solving the equation accurately, we find two solutions:
[tex]\[
x \approx 1 \quad \text{and} \quad x \approx 4.2
\][/tex]
4. Verification:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
(1)^2 - 5(1) + 6 = 1 - 5 + 6 = 2
\][/tex]
[tex]\[
\sqrt{1 + 3} = \sqrt{4} = 2
\][/tex]
Both sides match, so [tex]\( x = 1 \)[/tex] is a solution.
- For [tex]\( x \approx 4.2 \)[/tex]:
[tex]\[
(4.2)^2 - 5(4.2) + 6 \approx 17.64 - 21 + 6 = 2.64
\][/tex]
[tex]\[
\sqrt{4.2 + 3} = \sqrt{7.2} \approx 2.68
\][/tex]
These values are approximately equal, confirming [tex]\( x \approx 4.2 \)[/tex] as a solution.
5. Choice Selection:
Given the approximate solutions we have found, we look at the options provided:
[tex]\[
\text{C. } x = 1 \text{ and } x \approx 4.2
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
