Problem 6 - A Gentle Introduction to the Art of Mathematics

Question

Prove that for all real numbers \(a, b,\) and \(c,\) if \(a c<0,\) then the quadratic equation \(a x^{2}+b x+c=0\) has two real solutions.

Step-by-Step Solution

Answer

If \(ac < 0\), then \(b^2 - 4ac > 0\). Thus, the quadratic equation has two real solutions.

Step 1: Identify the Given Information

Given the quadratic equation: \(ax^2 + bx + c = 0\). It is also given that \(a c < 0\).

Step 2: Recall the Quadratic Formula

The solutions to the quadratic equation \(ax^2 + bx + c = 0\) are given by the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 3: Examine the Discriminant

The discriminant of the quadratic equation \(ax^2 + bx + c = 0\) is \(\Delta = b^2 - 4ac\). A quadratic equation has two real solutions if and only if \(\Delta > 0\).

Step 4: Analyze the Sign of the Discriminant

Since \(a\) and \(c\) have opposite signs (i.e., \(ac < 0\)), the term \(-4ac\) will be positive. Thus, \(b^2 - 4ac\) will be greater than \(b^2\).

Step 5: Conclude that the Discriminant is Positive

Because \(b^2 - 4ac > b^2\) and \(b^2\) is non-negative, it follows that \(\Delta > 0\). Therefore, the quadratic equation \(ax^2 + bx + c = 0\) has two real solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool in algebra for solving quadratic equations. It provides a straightforward method to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Here's what each part means:

• \(x\) - The solutions or roots of the quadratic equation.
• \(a, b, c\) - Coefficients from the equation where \(a \) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.
• \(\sqrt{b^2 - 4ac} \) - This part is called the discriminant and plays a key role in determining the nature of the roots.

The quadratic formula works universally for all quadratic equations, making it an essential method for students to master. By applying this formula, you can determine the roots quickly and efficiently.

Discriminant

The discriminant of a quadratic equation holds significant information about the nature of its roots. It is found within the quadratic formula and is represented by \(\Delta = b^2 - 4ac\)

Here are the possibilities concerning the discriminant's value:

• \(\Delta > 0\) - The quadratic equation has two distinct real solutions.
• \(\Delta = 0\) - The quadratic equation has exactly one real solution (a repeated root).
• \(\Delta < 0\) - The quadratic equation has no real solutions, but two complex solutions.

In our case, we are given that \(ac < 0\) which implies \(a\) and \(c\) have opposite signs. This makes the term \(-4ac\) positive, ensuring that \(\Delta > 0\). As a result, the quadratic equation \(ax^2 + bx + c = 0\) has two distinct real solutions. This positive discriminant tells us that the parabola intersects the x-axis at two points.

Real Solutions

A real solution in the context of quadratic equations refers to the x-values for which the quadratic equation equals zero. These solutions are also known as roots, and they can be found using the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The nature and number of real solutions depend on the discriminant, \(\Delta = b^2 - 4ac\). Since \(\Delta\) represents the part of the formula under the square root, its value dictates whether the solutions are real or complex:

• When \(\Delta > 0\), there are two distinct real solutions.
• When \(\Delta = 0\), there is exactly one real solution.
• If \(\Delta < 0\), there are no real solutions, only complex ones.

In scenarios where \(a c < 0\), we infer \(\Delta = b^2 - 4ac > 0\), meaning the quadratic equation will always have two distinct real solutions. This insight into the discriminant helps us understand the conditions necessary for quadratic equations to have real solutions.