Sure, let's solve the equation [tex]\(3x^2 = 7x - 3\)[/tex] step by step to find the solutions.
1. Rewrite the Equation:
First, we'll start by rewriting the given quadratic equation in standard form:
[tex]\[
3x^2 - 7x + 3 = 0
\][/tex]
2. Identify the Coefficients:
A quadratic equation in standard form is [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case:
[tex]\[
a = 3, \quad b = -7, \quad c = 3
\][/tex]
3. Apply the Quadratic Formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Plug in the identified coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}
\][/tex]
4. Simplify Inside the Square Root:
Calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[
\Delta = b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 3 = 49 - 36 = 13
\][/tex]
5. Complete the Formula:
Substitute the discriminant ([tex]\(\sqrt{13}\)[/tex]) back into the formula:
[tex]\[
x = \frac{7 \pm \sqrt{13}}{6}
\][/tex]
6. Separate the Solutions:
The solutions are:
[tex]\[
x = \frac{7 + \sqrt{13}}{6} \quad \text{and} \quad x = \frac{7 - \sqrt{13}}{6}
\][/tex]
Thus, the two solutions to the equation [tex]\(3x^2 = 7x - 3\)[/tex] are:
[tex]\[
x = \frac{7}{6} + \frac{\sqrt{13}}{6} \quad \text{and} \quad x = \frac{7}{6} - \frac{\sqrt{13}}{6}
\][/tex]
To summarize, the solutions to the quadratic equation [tex]\(3x^2 = 7x - 3\)[/tex] are:
[tex]\[
\left[\frac{7}{6} - \frac{\sqrt{13}}{6}, \frac{7}{6} + \frac{\sqrt{13}}{6}\right]
\][/tex]
This matches the given result exactly.
