Problem 92 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The solutions of \(3 x^{2}-5=0\) are \(\frac{\sqrt{5}}{3}\) and \(-\frac{\sqrt{5}}{3}\)

Step-by-Step Solution

Answer

The statement is false. The correct solutions to the given equation are \(x=\sqrt{\frac{5}{3}}\) and \(x=-\sqrt{\frac{5}{3}}\)

Step 1: Insert the Suggested Solutions

Begin by inserting the proposed solutions \(\frac{\sqrt{5}}{3}\) and \(-\frac{\sqrt{5}}{3}\) into the given equation \(3 x^{2}-5=0\) to assess their validity.

Step 2: Evaluate the Suggested Solutions

It's seen when substituting \(\frac{\sqrt{5}}{3}\) into the equation, the left side equates to \(-\frac{2}{3}\) and when substituting \(-\frac{\sqrt{5}}{3}\), the result is the same - neither equates to 0. Therefore, the given solutions are incorrect.

Step 3: Solve for the Correct Solution

Solve for \(x\) from the given equation \(3 x^{2}-5=0\). First, re-arrange the equation to get \(3x^2=5\) and then, \(x^2=\frac{5}{3}\). By taking the square root of each side, the correct solutions are found to be \(x=\sqrt{\frac{5}{3}}\) and \(x=-\sqrt{\frac{5}{3}}\)

Key Concepts

Quadratic Formula

Understanding the quadratic formula is essential when solving quadratic equations. Quadratic equations are algebraic expressions that are set to zero and follow the degree of 2, represented in the general form of \(ax^2 + bx + c = 0\). When factors are not easily identifiable or the equation cannot be easily simplified by factoring, the quadratic formula comes into play. This formula is \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\), where \(a\), \(b\), and \(c\) correspond to the coefficients and constant term in the equation.

Using the quadratic formula can sometimes seem daunting, but it's a reliable method because it always works for any quadratic equation. When applying this formula, one calculates the value under the square root (the discriminant), which determines the number of solutions. If the discriminant is positive, two real solutions exist; if it's zero, there is one real solution; and if negative, there are two complex solutions.

Algebraic Expressions

Algebraic expressions are combinations of letters (variables) and numbers (constants) using the operations of addition, subtraction, multiplication, division, and exponentiation. An equation like \(3x^2 - 5 = 0\) is an example of an algebraic expression set to zero, which makes it an equation to be solved.

Why Do Algebraic Expressions Matter?

They are the foundation of algebra and allow us to describe relationships and changes in relationships in a symbolic form. When it comes to solving them, especially quadratic equations, one must manipulate these expressions while respecting the arithmetic properties such as the distributive, associative, and commutative properties.

Improving our understanding of algebraic expressions also aids in recognizing patterns, predicting outcomes, and in developing advanced problem-solving skills, which are applicable in various scientific and engineering domains.

Square Roots

The square root operation is fundamental when working with quadratic equations. It is essentially the inverse operation of squaring a number. The square root of \(x\) is a number that, when multiplied by itself, will result in \(x\). For example, since \(2 * 2 = 4\), we say that \(2\) is the square root of \(4\), often written as \(\sqrt{4} = 2\).

Whenever you solve a quadratic equation by extracting square roots, it is crucial to consider both the positive and the negative square roots, as seen in the corrective steps of our original problem. This oversight can lead to incorrect or incomplete solutions. For instance, in the given equation \(3x^2=5\), taking the square root on both sides gives us \(\sqrt{3x^2} = \sqrt{5}\), leading to \(x = \pm\sqrt{\frac{5}{3}}\), rather than just the positive root, ensuring that we consider both possible values for \(x\) that satisfy the original equation.