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Factoring is a popular method for solving quadratic equations as it allows you to break down a complex equation into simpler parts. However, not all quadratic equations can be easily factored with integer solutions.
To factor a quadratic expression like \(ax^2 + bx + c\), you look for two numbers that multiply to \(ac\) and add to \(b\). The original equation can be split into these factors to solve for \(x\).
If we were able to factor \(9x^2 + 9x + 2\), we would express it in the form \((px + q)(rx + s) = 0\). Nevertheless, given how the numbers play out here, we find that this specific equation does not factor easily. That’s when it’s wise to look at other solving methods, like the quadratic formula.

The quadratic formula is a universal tool for solving quadratic equations. It is particularly useful when factoring is difficult or impossible with simple integers. The formula is expressed as follows:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This equation provides the solutions for any quadratic equation \(ax^2 + bx + c = 0\) by identifying the coefficients \(a\), \(b\), and \(c\).
To use the quadratic formula correctly, remember:

• Ensure the equation is set to zero.
• Identify the coefficients \(a\), \(b\), and \(c\).
• Substitute the coefficients into the formula.
• Calculate the discriminant \(b^2 - 4ac\) to see if roots are real or complex.

In our example \(9x^2 + 9x + 2 = 0\): we substitute \(a = 9\), \(b = 9\), \(c = 2\) into the formula to find the solutions. The quadratic formula is a great assurance that, even without factoring, we can still solve for the roots.

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). They can be found through factoring, using the quadratic formula, or graphically by finding where the parabola intersects the \(x\)-axis.
In the equation \(9x^2 + 9x + 2 = 0\), we found the roots using the quadratic formula, yielding \(x = -\frac{1}{3}\) and \(x = -\frac{2}{3}\). These are the points where the curve of the quadratic equation will touch or intersect the \(x\)-axis.
Here are some characteristics of quadratic roots:

• If the discriminant \(b^2 - 4ac\) is positive, there are two real distinct roots.
• If the discriminant equals zero, there is one real repeated root.
• If the discriminant is negative, the roots are complex and not real numbers.

Understanding the roots helps in interpreting the behavior of the quadratic function, including its direction and how it affects the graph of the equation.