Problem 24 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. $$-x^{2}+3 x=-4$$

Step-by-Step Solution

Answer

To solve the equation, first put it into standard form, then graph it. Identify where the graph intersects with the x-axis, as these points are the solutions. Validate the solutions by substitifying them back into the original equation.

Step 1: Write the Equation in Standard Form

The given equation is \(-x^2 + 3x = -4\). Let's write this equation in the standard form of a quadratic equation. Add 4 to both sides of the equation to obtain \( -x^2 + 3x + 4 = 0 \).

Step 2: Plot the Equation

Plot the given quadratic function \(y = -x^2 + 3x + 4\). The x values where the graph crosses the x-axis are the solutions of the equation.

Step 3: Estimate the Solutions by Analyzing the Graph

By analyzing the graph, estimate the x-values where the function intersects the x-axis. These are the solutions to the equation.

Step 4: Check the Found Solutions Algebraically

Substitute the estimated solutions from step 3 into the original equation and verify if the left side equals to right side (0). If it does, then those are the correct solutions.

Key Concepts

Graphing Quadratics

Graphing a quadratic equation involves plotting its parabola shape on a coordinate grid. In our example, the quadratic function is represented as \[ y = -x^2 + 3x + 4 \] To graph this equation, it's helpful to identify several key elements of the parabola:

• Vertex: The highest or lowest point on the graph depending on the leading coefficient (positive for the lowest, negative for the highest).
• Axis of symmetry: A vertical line that divides the parabola into two mirror images. It can be found using \( x = -\frac{b}{2a} \) when the equation is in standard form \( ax^2 + bx + c \).
• x-intercepts: These are the points where the graph crosses the x-axis and are essential for finding the solutions of the quadratic equation.

With this function, the negative coefficient in front of \( x^2 \) tells us that the parabola opens downward. By plotting several points, we will see where the curve crosses the x-axis to estimate the solutions of our equation. In this case, the solutions are the x-values where \( y \) becomes zero (i.e., where the graph intersects the x-axis). Adjusting the scale and making accurate plots ensures a better estimation of these points.

Solution Estimation

Once you have graphed the quadratic equation, the next step is estimating the solutions, which are the x-values of the points where the parabola intersects the x-axis. These x-values are the roots of the equation.
To do a solution estimation from the graph, follow these basic steps:

• Identify the approximate points where the parabola crosses the x-axis.
• Read off the x-coordinates of these points as the estimates of the solutions.

Keep in mind, these are estimated solutions. Because graphs may not be drawn to exact scale, multiple techniques including using a ruler or graphing technology can help improve accuracy. Once estimated, these x-values should logically make sense when plugging back into the context of the problem, but they are approximations.

Algebraic Verification

After estimating the solutions from the graph, it's important to verify these solutions algebraically. This process ensures the estimates are precise and mathematically correct. Here's how to check the solutions for \( -x^2 + 3x + 4 = 0 \).

• Take each estimated solution and substitute it back into the quadratic equation.
• Calculate the left and right side of the equation for this value.
• If both sides equal, the solution is correct; if not, re-evaluate your estimation.

This checks the true roots of the equation, providing confidence in the solutions found from the graph. It accounts for any graphical errors and provides the precise values mathematically. Remember, accurate algebraic verification confirms that these are actual x-intercepts and solutions of the quadratic equation.