Problem 40 - Algebra Form and Function

Question

For what values of \(a\) does the equation have a solution in \(x\) ? $$ x^{2}-a=0 $$

Step-by-Step Solution

Answer

Answer: The equation \(x^2 - a = 0\) has a real solution in \(x\) for all \(a\) greater than or equal to 0 (i.e., \(a \ge 0\)).

Step 1: Rearrange the Equation

We are given the equation: $$ x^2 - a = 0 $$ Add \(a\) to both sides to get: $$ x^2 = a $$

Step 2: Analyze the Discriminant

Now, let's analyze the discriminant. The discriminant is the part inside the square root of the quadratic formula, which is given by the expression \(\Delta = b^2-4ac\). In our case, the quadratic equation can be represented as \(x^2 - 0x - a = 0\). Comparing the equation with the general form of a quadratic equation, \(ax^2 + bx + c = 0\), we deduce that: - \(a = 1\) - \(b = 0\) - \(c = -a\) (from the given equation) Now, let's calculate the discriminant: $$ \Delta = b^2 - 4ac = (0)^2 - 4(1)(-a) $$ $$ \Delta = 4a $$

Step 3: Set Constraints for Real Solutions

For a quadratic equation to have real solutions, its discriminant must be greater than or equal to zero, i.e. \(\Delta \ge 0\). In our case: $$ 4a \ge 0 $$ Now, divide both sides by 4: $$ a \ge 0 $$ So, the equation \(x^2 - a = 0\) has a real solution in \(x\) for all \(a\) greater than or equal to 0.

Key Concepts

Discriminant

The concept of the discriminant is crucial when solving quadratic equations. It helps to determine the nature of the roots without solving the equation entirely. The discriminant is the part of the quadratic formula under the square root symbol, represented as \(\Delta = b^2 - 4ac\). Based on the value of the discriminant, we can easily find out how many and what kind of solutions an equation might have.

Here’s a simple breakdown:

• If \(\Delta > 0\), the quadratic equation has two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution, also known as a repeated or double root.
• If \(\Delta < 0\), the solutions are not real numbers but complex or imaginary.

In the exercise at hand, the discriminant was \(\Delta = 4a\). This tells us that for our specific equation, the crucial factor is the value of \(a\), as it determines whether the discriminant is positive, zero, or negative. This insight is what guides us in finding the conditions for which the equation has real solutions.

Real Solutions

When dealing with quadratic equations, it's important to distinguish between real and non-real solutions. Real solutions are simply the x-values that satisfy the equation and lie on the real number line. They help us make sense of problems occurring in real-world settings.

In the case of the equation \(x^2 - a = 0\), we want to find the conditions under which the solutions are real. From Step 3 of our solution, we found that the discriminant, \(\Delta = 4a\), must be greater than or equal to zero for the solutions to be real. Solving \(4a \geq 0\) gives us \(a \geq 0\).

This means that for non-negative values of \(a\), the equation will always have real solutions. These solutions correspond to points where the graph of \(y = x^2\) intersects with the line \(y = a\). When considering applications or interpreting data, having real solutions allows for tangible insights and predictions.

Algebra

Algebra is a field of mathematics that allows us to manipulate symbols and numbers to solve problems. It involves understanding and using elements such as variables, constants, and coefficients in equations. Quadratic equations like \(x^2 - a = 0\) are common in algebra.

To solve these equations, we often rely on algebraic methods such as factoring, using the quadratic formula, or simplifying expressions. For example, in the original exercise, we rearranged the equation to \(x^2 = a\). This transformation is a simple algebraic tool that sets the stage for further analysis to determine the values of \(x\) that satisfy the equation.

Learning algebra involves grasping fundamental concepts:

• Understanding the relationship between variables, constants, and coefficients.
• Applying rules and procedures to perform operations such as addition, subtraction, multiplication, and division on algebraic expressions.
• Utilizing methods to decipher equations and find solutions.

Mastering these mathematical tools provides a foundation for understanding and solving more complex problems, both in academic contexts and real-life situations.