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Breaking down a quadratic equation into simpler factors is called factoring. Let’s apply this concept to the equation provided: When you first look at the equation, it appears as: \[3x^2 + 3x - 6 = 0\]To make this task of factoring easier, dividing the entire equation by 3 simplifies it:\[x^2 + x - 2 = 0\]Now, the task is to find two numbers whose product is -2 (the constant term) and whose sum is 1 (the coefficient of x). If we consider the numbers -1 and 2, they fit the bill perfectly: - Sum: \( -1 + 2 = 1 \) - Product: \( -1 \times 2 = -2 \) Using these numbers, the quadratic expression is factored nicely as:\[(x - 1)(x + 2) = 0\]Thus factoring reduces the complexity of solving equations and helps discover the roots swiftly.

Roots are solutions of the quadratic equation that make it equal to zero. Each factor of a factored quadratic equation can help identify a root. From the factored form of our equation:\[(x - 1)(x + 2) = 0\]You acquire two separate "mini-equations" by setting each factor to zero:- First, set \(x - 1 = 0 \) which solves to \(x = 1 \). - Second, set \(x + 2 = 0 \) which leads to \(x = -2 \).These values, 1 and -2, are the roots of the quadratic equation. They are also referred to as solutions or x-intercepts. They reveal the points where the parabola, representing the quadratic, crosses the x-axis. Finding roots is crucial as it tells us where the function equals zero, thus serving varied purposes in graphing and real-world problems.

The graphical representation of a quadratic equation is a parabola. In this instance, the equation \(3x^{2} + 3x = 6\) transforms into the parabola after solving for y:\[y = x^2 + x - 2\]Using the roots found (1 and -2), these points, (1,0) and (-2,0), are where the parabola crosses the x-axis—known as x-intercepts.To sketch this parabola further, locate the y-intercept by setting \(x = 0\):\[y = 0^2 + 0 - 2 = -2\]This gives us another point, (0,-2), where the parabola crosses the y-axis.With these three key points: - (1, 0)- (-2, 0)- (0, -2)We construct a general sketch of the parabola. The parabola curves upwards as the leading coefficient (before \(x^2\)) is positive, \ indicating a minimum point. Visualizing the graph helps in understanding the behavior and solution set of the equation.