Problem 42 - Algebra 1: Concepts and Skills

Question

Solve the equation algebraically. Check your solutions by graphing. $$2 x^{2}-89=9$$

Step-by-Step Solution

Answer

The solutions to the equation are x = 7 and x = -7

Step 1: Simplify the Quadratic Equation

The given equation is \(2x^2 - 89 = 9\). First, let's reorganize the equation so that all the terms are on one side. Add 89 to both sides to form a simplified equation: \(2x^2 = 98\).

Step 2: Solve for x

Next, let's solve for x. Divide each side of the equation by 2, making the equation \(x^2 = 49\). The square root of both sides give us the solutions \(x = 7\) and \(x = -7\)

Step 3: Graph and Check Solution

To confirm if the solutions are correct, graph the equation \(y = 2x^2 - 89\). The graph should intersect the x-axis at points x = 7 and x = -7, confirming that these are the correct solutions

Key Concepts

Quadratic Formula

A quadratic equation typically has the form \(ax^2 + bx + c = 0\). The quadratic formula is one of the most powerful tools to solve any quadratic equation, given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps find the roots, or solutions, of the quadratic equation. It is derived from completing the square method. The expression under the square root sign, \(b^2 - 4ac\), is called the discriminant. It tells us the nature of the roots. If it is positive, there are two different real roots. If zero, one real root exists. A negative value indicates no real roots, but rather complex ones. Applying the quadratic formula is a precise algebraic method, unlike graphical solutions that visually estimate roots.

Graphical Solution

Graphing is an excellent way to verify solutions of a quadratic equation. By converting the standard form equation, \(ax^2 + bx + c = 0\), to a function \(y = ax^2 + bx + c\), we can draw its graph on a coordinate plane.
The points where the graph intersects the x-axis represent the solutions of the equation. In our example, graphing \(y = 2x^2 - 89\) shows intersections at \(x = 7\) and \(x = -7\), confirming our solutions. Using graphing calculators or software makes this process much simpler.

• Ensure the graph's accuracy by checking the vertex and axis of symmetry.
• Check for correct intersection points that match algebraic solutions.

Graphical methods can provide a visual understanding of the equation's behavior and help in grasping the concept of the roots more intuitively.

Simplifying Equations

Simplifying equations is a crucial step in solving a quadratic equation efficiently. The initial equation \(2x^2 - 89 = 9\) was first reorganized to \(2x^2 = 98\).
This involves shifting all terms to one side to have a zero on the other, making it ready for factoring or applying formulas. Next, we divided through by 2, resulting in \(x^2 = 49\). This step is crucial to isolate the variable terms, thus simplifying the equation further.

• Perform arithmetic operations to eliminate terms.
• Factor out common factors or use algebraic identities if possible.

Once simplified, solving becomes straightforward, either through taking square roots, factoring, or applying the quadratic formula. Mastery of simplification techniques aids in solving not only quadratic equations but a wide array of algebraic problems.