Answer:
The solutions for the equation \(3x^{2} =7x-3\) are: \(x_{1}=\frac{7+\sqrt{13}}{6} , x_{2}= \frac{7-\sqrt{13} }{6}\), so the option d is correct.
Step-by-step explanation:
You have two options to find the solutions of the equation \(3x^{2} =7x-3\)
The first is to solve it with the quadratic formula, for this, you need to follow these steps:
- Add 3 and substract 7x from both sides \(3x^{2} +3-7x=7x+3-3-7x\)
- Simplify \(3x^{2} +3-7x=0\)
- Use the quadratic equation formula. For a quadratic equation of the form \(ax^{2} +bx+c=0\) the solutions are \(x_{1,2} = \frac{-b\±\sqrt{b^{2}-4ac } }{2a}\)
- In the equation given \(a=3, b=-7, c=3\), so \(x_{1} = \frac{-(-7)+\sqrt{(-7)^{2}-4*3*3 } }{2*3}, x_{2} = \frac{-(-7)-\sqrt{(-7)^{2}-4*3*3 } }{2*3}\)
- The solutions are \(x_{1}=\frac{7+\sqrt{13}}{6} , x_{2}= \frac{7-\sqrt{13} }{6}\)
The second option is solve the equation by completing the square:
- Substract 7x from both sides \(3x^{2} -7x=7x-7x-3\) and simplify \(3x^{2} -7x=-3\)
- Divide both sides by 3 \(\frac{3x^{2}-7x }{3}= -\frac{3}{3}\) and simplify \(x^{2} -\frac{7x}{3}=-1\)
- Use the fact that \(x^{2} +2ax+a^{2}=(x+a)^{2}\)
- We need to find \(a\), for this, we use this relation \(2ax=-\frac{7x}{3}\) and solve for \(a\), we get \(a = -\frac{7}{6}\)
- Add \(a^{2} = (\frac{-7}{6} )^{2} =\frac{13}{36}\) to both sides in the equation of step 2 \(x^{2} -\frac{7x}{3}+(-\frac{7}{6} )^{2}=-1 +(-\frac{7}{6} )^{2}\) and simplify \(x^{2} -\frac{7x}{3}+(-\frac{7}{6} )^{2}=\frac{13}{36}\)
- Complete the square, use step 3, \(x^{2} -\frac{7x}{3}+(-\frac{7}{6} )^{2} = (x-\frac{7}{6} )^{2}=\frac{13}{36}\)
- For \(f^{2}(x)\) the solutions are \(f(x) =\sqrt{a},-\sqrt{a}\), so \(x-\frac{7}{6} = \sqrt{\frac{13}{36} } =\frac{\sqrt{13}+7 }{6}\ and \ x-\frac{7}{6} = -\sqrt{\frac{13}{36} } =-\frac{\sqrt{13}+7 }{6}\)
