Problem 32 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. $$2 x^{2}-8 x=10$$

Step-by-Step Solution

Answer

The solutions of the equation \(2 x^{2}-8 x=10\) can be estimated from the graph and should be checked by substituting them back into the original equation.

Step 1: Rewrite the equation

The first step is to re-write the equation for y. Do this by subtracting 10 from both sides of the equation. It becomes: \(2x^2 - 8x - 10 = 0\)

Step 2: Graph the equation

Graph \(y = 2x^2 - 8x - 10\). Identify the x-intercepts – these are the solutions of the equation. The x-intercepts can be found where the graph crosses the x-axis (where y=0).

Step 3: Estimate the solutions

Estimate the solutions from where the graph crosses the x-axis. Should observe two values where graph intersects x-axis providing us with two solutions.

Step 4: Check the solutions algebraically

Substitute the estimated solutions back into the original equation \(2 x^{2}-8 x=10\) and see if left hand side equals to the right hand side. If they do, then the solutions are correct.

Key Concepts

Graphing Quadratics

Graphing a quadratic equation is a visual method to find its solutions, providing a clear picture of the values we are interested in. When we deal with quadratics, they typically take a parabolic form, meaning they curve outward or inward like a smooth "U" shape. In our exercise, the quadratic function is given by:

\[ y = 2x^2 - 8x - 10 \]This equation represents a parabola. To graph this, you can start by creating a table of values, choosing a range of x values, and calculating their corresponding y values. By plotting these pairs on the graph and connecting them smoothly, you'll form a parabola.

Finding the solutions of this equation is about discovering where this parabola crosses the x-axis. These crossing points are the x-intercepts, which give us the roots or solutions of the quadratic equation. Graphing provides a visualization of solutions, showing where the equation equals zero.

X-Intercepts

In the context of quadratic functions, the x-intercepts are where the graph touches or crosses the x-axis. These points occur where the output, or y-value, is zero. For a quadratic equation in standard form, such as:

\[ y = ax^2 + bx + c \]The x-intercepts are solutions to the equation \( ax^2 + bx + c = 0 \). In our exercise, we found these intercepts by rerouting the equation to form:

\[ 2x^2 - 8x - 10 = 0 \]Plotting this equation helps identify where the parabola intersects the x-axis. Each point of intersection is a potential solution to the equation. In the graphically solved example, you would observe these intersections and estimate their values. Reporting these points allows us to identify our potential solutions visually.

Algebraic Solutions

Once we've gathered potential solutions from graphing, it's time to verify them algebraically. Algebraic methods give precise verification for what the graph estimates. For our equation:

\[ 2x^2 - 8x - 10 = 0 \]A common method to solve it involves factoring, using the quadratic formula, or even completing the square. In this case, the purpose is to check if the x values we estimated from the graph are indeed solutions.

To verify algebraically, substitute the guessed x-values back into the original equation and ensure that both sides of the equation equal. If substituting a value yields a true equation, then that x-value is an accurate solution. Algebraic verification is a critical step because it confirms the graphical estimates with exact numbers, ensuring accuracy in our results.