To determine how many real solutions the equation [tex]\(\sqrt{3x - 1} = 2x - 5\)[/tex] has, we need to analyze and solve the equation step-by-step.
First, let's rewrite the equation:
[tex]\[
\sqrt{3x - 1} = 2x - 5
\][/tex]
To eliminate the square root, we square both sides of the equation:
[tex]\[
(\sqrt{3x - 1})^2 = (2x - 5)^2
\][/tex]
This simplifies to:
[tex]\[
3x - 1 = (2x - 5)^2
\][/tex]
Next, we expand the right-hand side:
[tex]\[
3x - 1 = 4x^2 - 20x + 25
\][/tex]
Now, we bring all terms to one side of the equation to set it to zero:
[tex]\[
4x^2 - 20x + 25 - 3x + 1 = 0
\][/tex]
Combining like terms, we get:
[tex]\[
4x^2 - 23x + 26 = 0
\][/tex]
This is a quadratic equation in standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 4\)[/tex], [tex]\(b = -23\)[/tex], and [tex]\(c = 26\)[/tex].
To determine the number of real solutions, we use the discriminant [tex]\(\Delta\)[/tex] of the quadratic equation, given by [tex]\(\Delta = b^2 - 4ac\)[/tex].
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-23)^2 - 4 \cdot 4 \cdot 26 = 529 - 416 = 113
\][/tex]
Since the discriminant [tex]\(\Delta > 0\)[/tex], there are two distinct real roots for the quadratic equation.
However, since these roots come from squaring both sides of an equation, we must check if they are valid solutions to the original equation. After performing this check, we find that only one of these roots satisfies the original equation.
Thus, the equation [tex]\(\sqrt{3x - 1} = 2x - 5\)[/tex] has exactly one real solution.
The correct number of real solutions is:
B. 1
