Problem 32 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The solutions to the equation \(2x^2 - 89 = 9\) are \(x = 7\) and \(x = -7\).

Step 1: Rearrange the Equation

First, rearrange the equation into the form \(ax^2 + bx + c = 0\). To do this, add 89 to both sides of the equation and subtract 9 from both sides to obtain the equation \(2x^2 - 98 = 0\).

Step 2: Apply the Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) can now be used to solve for \(x\). Applying the quadratic formula, we get \(x = \frac{-0 \pm \sqrt{0 - 4(2)(-98)}}{2(2)} = \frac{\pm \sqrt{784}}{4}\), yielding the solutions \(x = 7\) and \(x = -7\).

Step 3: Graphically Check the Solutions

Make a graph of the function \(f(x) = 2x^2 - 89\) and verify that it intersects the x-axis (i.e., that \(f(x) = 0\)) at \(x = 7\) and \(x = -7\).

Key Concepts

Quadratic Formula

The quadratic formula, an essential tool in algebra, swiftly finds the roots of a quadratic equation, which is any equation having the form \(ax^2 + bx + c = 0\). Its beauty lies in its ability to provide solutions for \(x\) regardless of whether the quadratic trinomial factors nicely. Given the coefficients \(a\), \(b\), and \(c\), the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) calculates the solutions directly.

When applying the quadratic formula, it's crucial to consider the term under the square root, \(b^2 - 4ac\), known as the discriminant. This term determines the nature of the roots:

• If \(b^2 - 4ac > 0\), the equation has two real and distinct solutions.
• If \(b^2 - 4ac = 0\), the equation has exactly one real solution (also called a repeated or double root).
• If \(b^2 - 4ac < 0\), the equation has no real solutions, implying the roots are complex numbers.

Understanding and utilizing the quadratic formula facilitates the handling of equations that are not easily factorable, providing a systematic approach to find real or complex solutions.

Algebraic Solutions

Algebraic solutions to quadratic equations involve manipulating the equation to isolate \(x\) and find its values. The traditional strategy commences with bringing the equation to standard form, \(ax^2 + bx + c = 0\), to prepare for factoring or applying the quadratic formula. Factoring, an alternative to the quadratic formula, involves identifying two binomials whose product gives the original quadratic trinomial. However, factoring demands the trinomial to break down neatly into such factors, which isn't always possible.

For a quadratic that is not easily factorable, the quadratic formula prevails as the most reliable method. After identifying the coefficients and substituting them into the formula, the algebraic solution is a matter of carrying out arithmetic operations. The ability to manipulate and simplify expressions underpins the successful application of algebraic methods. When students encounter a quadratic equation, grasping these algebraic techniques equips them to determine the roots systematically, even when a graphical solution isn't readily available.

Graphical Solutions

Graphical solutions offer a visual interpretation of quadratic equations by turning the algebraic expression into a parabolic graph. This graph provides a geometric perspective of the solutions, which are the points where the parabola intersects the x-axis. Plotting the function \(f(x) = ax^2 + bx + c\) and observing its shape and intersection points yields insights into the behavior of the quadratic function.

A parabola can open upwards or downwards, dictated by the sign of the leading coefficient \(a\). When graphing, the vertex represents the peak or trough of the parabola and the axis of symmetry runs vertically through it. Additionally, the roots depicted on the graph correspond to the x-values found algebraically, thus offering confirmation of the calculated solutions. By understanding the relationship between algebraic and graphical solutions, students gain a more comprehensive grasp of quadratic functions, enhancing their problem-solving skills in both concrete and abstract contexts.