To solve the given quadratic equation, we need to factorize the given expression.
To do it, the first step is to multiply all terms in the equation by the coefficient of the first term (7):
[tex]\begin{gathered} 7\cdot(7x^2-31x+12)=0 \\ 49x^2-217x+84=0 \end{gathered}[/tex]Now, re write the equation in terms of 7x:
[tex]\begin{gathered} 49x^2-217x+84=0 \\ (7x)^2-31(7x)+84=0 \end{gathered}[/tex]Now, factorize the expression, look for 2 numbers whose product equals 84 and whose sum equals 31. Those numbers are 28 and 3.
[tex](7x-28)(7x-3)=0[/tex]Now, divide the expression by the coefficient of the first term in the original equation (7):
[tex]\begin{gathered} \frac{(7x-28)(7x-3)}{7}=0 \\ (x-4)(7x-3)=0 \end{gathered}[/tex]Now, for the expression to be equal to 0, one of the factors must be 0. Use this information to find the values of x:
[tex]\begin{gathered} x-4=0 \\ x=4 \\ 7x-3=0 \\ 7x=3 \\ x=\frac{3}{7} \end{gathered}[/tex]The solutions are x=4 and x=3/7.