Question

Solve the equation by graphing the related system of equations. $$4 x^2-3 x-6=-x^2+5 x+3$$

Short answer

The solutions to the equation are the x-coordinates of the points where the two parabola's on the plot intersect.

Step-by-step solution

Rewrite the equation into a system of equations

We can rewrite the given equation as two separate simpler equations:\n1) \(4 x^2 - 3x - 6 = 0\)\n2) \(-x^2 + 5x + 3 = 0\) These are the equations of two parabolas, which we can now graph.

Graph the equations

We graph the equations on the same plot. We'll see that they intersect at certain points.

Find the intersection points

The solutions to the original equation are the x-coordinates of the points where the two graphs intersect.

Key concepts

Graphical Solution

A graphical solution involves solving an equation by visually interpreting the graphs of the equations involved. In this method, we equate the given equation to zero by separating it into two different equations, often involving parabolas or lines. Then, we graph these equations on the same set of axes.

In the context of quadratic equations, each equation forms a parabolic curve when plotted. Observing these curves on the Cartesian plane allows us to identify the points where they intersect. These intersection points help us find the solutions to the equation we are solving. This approach is particularly useful because it provides a visual understanding and verification of the solutions.

When graphing, we commonly observe the following:

• The shape of the graph for quadratic functions is always a parabola.
• A parabola can open upwards or downwards depending on the sign of its leading coefficient.
• The intersection of these graphs gives us the solution to the problem.

Thus, the graphical solution not only helps to find the solution but also offers a visual depiction of the relationship between the equations.

Intersection Points

Intersection points on a graph are the coordinates where two or more graphs meet. For the problem at hand, these are crucial as they represent the solutions to our quadratic equation system.

Finding the intersection points involves:

• Graphing each equation to form a visual representation.
• Identifying where the curves cross each other on the graph.

Once these points are located, the x-values of these points indicate the solutions to the equation.

Knowing the role of intersection can also highlight the significance of symmetry in quadratic equations, as well as how shifts in the graph (vertical or horizontal) affect the intersection points. Furthermore, analyzing these points grants us deeper insight into the geometric representation of algebraic solutions.

System of Equations

A system of equations consists of multiple equations that are considered simultaneously. For this exercise, the system consists of two quadratic equations. We rewrite the original equation as two separate equations, each a standard form of a quadratic equation:

1) \(4x^2 - 3x - 6 = 0\)
2) \(-x^2 + 5x + 3 = 0\).

This transformation turns the original problem into a more manageable form. Instead of solving one complex equation, we deal with simpler individual equations.

Graphing these gives us an intersection view approach, as elucidated by the graphical solutions section.

• The goal is to find a common solution that satisfies both equations.
• These points of intersection directly relate to the solutions for the original problem.

Employing the system of equations approach can simplify complex problems into comprehensible components, paving the way for easier computation and understanding.