Math Problem Solver

Solve by factoring:

Factor the nonzero side of the equation completely. $$\begin{aligned} \left(9 x - 6\right)^{2} - 9 &= 0 \\ 27 \cdot \left(3 x - 1\right) \left(x - 1\right) &= 0\end{aligned}$$

The product of any number and zero is zero, so the equation will hold true if either of the factors is equal to zero. Set each of the factors equal to zero and solve. $$\begin{aligned} x - 1 &= 0\\ x &= 1\end{aligned}$$ $$\begin{aligned} 3 x - 1 &= 0\\ x &= \frac{1}{3}\end{aligned}$$

Solve by quadratic formula:

Given a quadratic equation \(a x^{2} + b x + c = 0\), where \(a\), \(b\), \(c\) are constants and \(a \ne 0\), the solutions are given by the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\] In this case \(a=81\), \(b=-108\), and \(c=27\).

The discriminant is the quantity under the square root sign in the quadratic formula, and its sign determines the number of solutions to the quadratic equation when the coefficients are real. The discriminant is:\[b^{2}-4ac = \left(-108\right)^{2} - 324 \cdot 27=2916 > 0\] The discriminant is greater than zero, so this quadratic equation has two real solutions.

The two solutions are: \[x = \frac{\left(-1\right) \left(-108\right) + \sqrt{2916}}{2 \cdot 81} = 1 \] \[x = \frac{\left(-1\right) \left(-108\right) - \sqrt{2916}}{2 \cdot 81} = \frac{1}{3} = 0.\overline{3}\]