Problem 42 - Algebra Form and Function

Question

For what values of \(a\) does the equation have a solution in \(x\) ? $$ a x^{2}-5=0 $$

Step-by-Step Solution

Answer

Answer: The quadratic equation \(ax^2 - 5 = 0\) has real solutions for x when \(a \ge 0\).

Step 1: Identify the constants in the equation

In the given quadratic equation, \(ax^2 - 5 = 0\), the constants are: - a is the coefficient of the \(x^2\) term - b is the coefficient of the x term (not given, therefore b = 0) - c is the constant term. In this case, c = -5.

Step 2: Calculate the discriminant

The discriminant, \(\Delta\), of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: $$ \Delta = b^2 - 4ac $$ Since b = 0 in our equation, we substitute the values of a and c into the formula: $$ \Delta = 0^2 - 4a(-5) $$ Simplifying, we get: $$ \Delta = 20a $$

Step 3: Determine the condition for real solutions

For a quadratic equation to have real solutions, the discriminant must be greater than or equal to 0. So, we will find the values of \(a\) for which \(\Delta \ge 0\). $$ 20a \ge 0 $$ Divide by 20: $$ a \ge 0 $$

Step 4: State the solution

The quadratic equation \(ax^2 - 5 = 0\) has real solutions for x when \(a \ge 0\).

Key Concepts

Discriminant

The discriminant is a key concept in understanding quadratic equations. It helps determine the nature and number of solutions an equation might have. In simple terms, the discriminant is a part of the quadratic formula that you can use to decide how many solutions a quadratic equation will have and whether they are real or complex.

For a quadratic equation in the form \[ax^2 + bx + c = 0,\]the discriminant (often represented by the symbol \( \Delta \)) is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

The value of \( \Delta \) can tell you a lot about the equation:

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

Understanding this concept allows us to identify not only how many solutions exist but also what type they are. This makes the discriminant an invaluable tool when studying quadratic equations.

Real Solutions

When dealing with quadratic equations, you will inevitably encounter the notion of real solutions. A real solution is a solution that can be represented on the number line; it is not imaginary or complex. To determine whether a quadratic equation has real solutions, the discriminant plays a critical role.

For instance, in our equation \( ax^2 - 5 = 0 \), we calculated the discriminant to be \( \Delta = 20a \). To ensure the equation has real solutions, we need this discriminant to fulfill the condition:

• \( \Delta \geq 0 \)

This condition ensures that the equation has at least one real solution:

• If \( \Delta > 0 \), then two distinct real solutions exist.
• If \( \Delta = 0 \), then a single real solution exists.

Thus, for our equation to have real solutions, \( a \) must be greater than or equal to 0, leading us to conclude that our equation has real solutions for \( a \geq 0 \). By understanding the role of the discriminant, you can confidently determine the reality of solutions for any quadratic equation.

Coefficient Identification

Identifying the coefficients in a quadratic equation is an initial but crucial step in solving it. Coefficients are the numbers that multiply the variables in an equation. For any quadratic equation of the form:\[ax^2 + bx + c = 0,\]understanding which terms represent \(a\), \(b\), and \(c\) is essential.

Let's break this down:

• \(a\) is the coefficient of the \(x^2\) term.
• \(b\) is the coefficient of the \(x\) term. If no \(x\) term is visible, like in our equation \(ax^2 - 5 = 0\), then \(b = 0\).
• \(c\) is the constant term, which in our problem is \(-5\).

By carefully identifying these coefficients, solving the quadratic becomes a more straightforward process. In the given example, once you know that \(a\) is the coefficient in front of \(x^2\) and \(c = -5\), you can easily apply these values to find the discriminant and solve the equation.