To solve the equation [tex]\(\sqrt{3x - 1} = 2x - 5\)[/tex] and determine how many real solutions it has, we follow these steps:
1. Square Both Sides: Start by squaring both sides of the equation to eliminate the square root. This gives us:
[tex]\[
(\sqrt{3x - 1})^2 = (2x - 5)^2
\][/tex]
[tex]\[
3x - 1 = (2x - 5)^2
\][/tex]
2. Expand the Squared Term on the Right Side:
[tex]\[
3x - 1 = (2x - 5)(2x - 5)
\][/tex]
[tex]\[
3x - 1 = 4x^2 - 20x + 25
\][/tex]
3. Rearrange the Equation:
Bring all terms to one side to set the equation to zero:
[tex]\[
0 = 4x^2 - 20x + 25 - 3x + 1
\][/tex]
[tex]\[
0 = 4x^2 - 23x + 26
\][/tex]
4. Solve the Quadratic Equation:
Next, we solve the quadratic equation [tex]\(4x^2 - 23x + 26 = 0\)[/tex] using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 4\)[/tex], [tex]\(b = -23\)[/tex], and [tex]\(c = 26\)[/tex].
[tex]\[
x = \frac{23 \pm \sqrt{(-23)^2 - 4 \cdot 4 \cdot 26}}{2 \cdot 4}
\][/tex]
[tex]\[
x = \frac{23 \pm \sqrt{529 - 416}}{8}
\][/tex]
[tex]\[
x = \frac{23 \pm \sqrt{113}}{8}
\][/tex]
This provides us with two potential solutions:
[tex]\[
x_1 = \frac{23 + \sqrt{113}}{8}
\][/tex]
[tex]\[
x_2 = \frac{23 - \sqrt{113}}{8}
\][/tex]
5. Check for Validity: Even though we have two solutions, we need to check if both are valid by substituting them back into the original equation [tex]\(\sqrt{3x - 1} = 2x - 5\)[/tex]:
- Substituting [tex]\(x_1 = \frac{23 + \sqrt{113}}{8}\)[/tex] back into the original equation to see if it satisfies the equation
- Substituting [tex]\(x_2 = \frac{23 - \sqrt{113}}{8}\)[/tex] back into the original equation to see if it satisfies the equation
Upon verification, we realize that among these solutions, only one satisfies the original equation. Hence, there is exactly one real solution.
Therefore, the number of real solutions to the equation [tex]\(\sqrt{3x - 1} = 2x - 5\)[/tex] is:
B. 1
