Problem 58 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. \(0=3 x^{2}-10 x-4\)

Step-by-Step Solution

Answer

The roots are between 2 and 3 and between -0.5 and 0.

Step 1: Understanding the Problem

The task is to find the roots of the quadratic equation \(0 = 3x^2 - 10x - 4\) by graphing. The roots are the values of \(x\) where the graph intersects the x-axis.

Step 2: Graph the Quadratic Equation

Graph the equation \(y = 3x^2 - 10x - 4\) on the coordinate plane. The parabola will open upwards because the coefficient of \(x^2\) is positive.

Step 3: Identify Intersection Points with X-axis

Observe the graph to see where it crosses the x-axis. These points are the roots of the equation.

Step 4: Determine Exact or Approximate Roots

By observing the graph, if the parabola crosses the x-axis at integer values, these are the exact roots. If not, identify the consecutive integers the roots fall between.

Key Concepts

Quadratic Equation

A quadratic equation is a special type of polynomial equation that is characterized by the highest power of the variable being a square, or 2. Its general form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). In the given exercise, the quadratic equation is \(3x^2 - 10x - 4 = 0\). Quadratic equations are very common in mathematics due to their simple yet powerful function type. The solutions or "roots" of these equations are the values of \(x\) that satisfy the equation and are of fundamental importance in various fields such as physics, engineering, and economics.

Graphing

Graphing a quadratic equation involves plotting it on a coordinate plane. To graph the equation \(y = 3x^2 - 10x - 4\), we need to recognize its shape and key features. The graph of any quadratic equation is a parabola. Begin by identifying the direction of the parabola, which, in this case, opens upwards because the coefficient of \(x^2\) is positive.
Next, calculate key points, such as the vertex and y-intercept. A useful method is to create a table of values by selecting various \(x\) values and calculating corresponding \(y\) values to plot their points. Finally, connect the points smoothly to reveal the parabolic shape. This visual representation helps us understand the behavior of the quadratic function and find its roots.

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. Depending on the coefficient of the quadratic term, a parabola can open upwards or downwards. For our quadratic function \(y = 3x^2 - 10x - 4\), it opens upwards because \(3 > 0\).
The vertex of the parabola, which can be found using the formula \(x = -\frac{b}{2a}\), gives us the highest or lowest point of the curve, depending on its orientation. The axis of symmetry runs vertically through the vertex, and the parabola is symmetric about this line. Understanding these properties helps in effectively sketching the graph and locating the intersection points with the x-axis, which are crucial for solving the equation by graphing.

Roots of Equations

The roots of a quadratic equation, also known as solutions or zeros, are the x-values where the quadratic polynomial equals zero. For the given quadratic equation, the task is to find these roots by looking at the graph of the function.
When graphed, the roots correspond to the points where the parabola intersects the x-axis. These intersection points give the values of \(x\) that satisfy the equation \(y=3x^2-10x-4=0\). Sometimes, these values are integers, making the roots exact. Other times, they may fall between two consecutive integers, offering an approximate solution when truly solving via graph requires more precision than a sketch allows. In our exercise, if the parabola doesn’t intersect at whole numbers, you should note the range between which the exact roots are found.

X-Intercepts

The x-intercepts of a parabola are essential in solving quadratic equations by graphing. These intercepts are where the parabola crosses or touches the x-axis, helping to visualize the equation's roots. It is important to visually inspect the graph to determine these intercepts accurately.
When the parabola crosses the x-axis, at these points, the y-value is zero, fulfilling the equation’s requirement \(ax^2 + bx + c = 0\). In our problem, finding these x-intercepts will directly give us the roots of the quadratic equation \(3x^2 - 10x - 4 = 0\). If you're unable to identify exact intercepts due to scaling or graph precision, stating the consecutive integers between which the x-intercepts are located can offer guidance towards refining your solution using other methods.