Problem 38 - Algebra 1

Question

Represent the solution graphically. Check the solution algebraically. $$ 3 x^{2}+3 x=6 $$

Step-by-Step Solution

Answer

The solutions for the equation \(3x^2 + 3x = 6\) are \(x = 1\) and \(x = -2\). The graphical representation is a parabola with roots at these points and the y-intercept at (0,-2). Algebraic verification confirms the accuracy of these solutions.

Step 1: Standardizing the equation

The formula given is \(3x^2 + 3x = 6\). To make it standard form, bring all the terms to one side of the equation. Subtract 6 from both sides of the equation to get the result as \(3x^2 + 3x - 6 = 0\)

Step 2: Factoring the equation

Factor the quadratic equation. First divide the whole equation by 3 to simplify it, which results to \(x^2 + x - 2 = 0\). Further factoring the left-hand side we get \((x-1)(x+2) = 0\).

Step 3: Finding the roots

Set each factor equal to zero and solve for x. Your solutions or roots are \(x = 1\) and \(x = -2\)

Step 4: Graphical Representation

The graphic of this quadratic equation will be a parabola. The roots found earlier represent the x-intercepts of this parabola. The y-intercept is found by setting x = 0, in which case y = -2. This yields three points that can be used to sketch the curve: (1,0), (-2,0) and (0,-2).

Step 5: Algebraic Verification

To check the solution algebraically, substitute the found roots into the original equation. Substituting 1 gives \(3*1^2 + 3*1 = 6\), which simplifies to 6=6, and it's true. Substituting -2 gives \(3*(-2)^2 + 3*(-2) = 6\), which also simplifies to 6=6, and it's true. Thus, the solutions are verified.

Key Concepts

Factoring

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 of a Quadratic Equation

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.

Graphical Representation

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.