Problem 183 - Calculus Volume 1

Question

Consider the roots of the equation. Find the conditions for exactly one root (double root) for the equation \(y=x^{2}+b x+c\)

Step-by-Step Solution

Answer

The condition for a double root is \(b^2 = 4c\).

Step 1: Identify the Characteristics of a Double Root

For a quadratic equation to have exactly one (double) root, the discriminant must be zero. The discriminant for the quadratic equation \(y = ax^2 + bx + c\) is given by \(\Delta = b^2 - 4ac\). Therefore, \(\Delta = 0\) ensures a double root.

Step 2: Plug Coefficients into Discriminant Formula

For the given equation \(y = x^2 + bx + c\), the coefficients are \(a = 1\), \(b = b\), and \(c = c\). Substitute these values into the discriminant formula: \(\Delta = b^2 - 4 \cdot 1 \cdot c\), which simplifies to \(\Delta = b^2 - 4c\).

Step 3: Set the Discriminant to Zero

To find the condition for a double root, set the discriminant to zero: \(b^2 - 4c = 0\). This gives us the equation of equality: \(b^2 = 4c\).

Step 4: Express the Condition Clearly

This equation \(b^2 = 4c\) is the condition required for the quadratic equation \(y = x^2 + bx + c\) to have exactly one root (a double root). It states that the square of the coefficient \(b\) must be equal to four times the constant term \(c\).

Key Concepts

Discriminant

In the world of quadratic equations, the discriminant is a crucial concept that helps us determine the nature of the roots of an equation. For a quadratic equation of the form \(y = ax^2 + bx + c\), the discriminant is represented by \(\Delta\) and is calculated using the formula \(\Delta = b^2 - 4ac\).

The value of the discriminant provides insight into how many and what type of roots the quadratic equation possesses:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, known as a double root.
• If \(\Delta < 0\), the roots are complex and not real.

Understanding the discriminant helps in quickly identifying the characteristics of the equation's roots. It is a powerful tool for analyzing quadratics without having to solve them completely.

Double Root

A double root, also known as a repeated root, occurs when a quadratic equation has exactly one real solution. When the discriminant \(\Delta\) is zero, \(\Delta = 0\), it signifies that the equation does not "open out" into two distinct roots but rather "closes in" on a single root.

In mathematical terms, this means both roots of the quadratic equation are the same. For example, in the equation \(y = (x - a)^2\), \(x = a\) is the double root.

Having a double root indicates a perfect square trinomial, and graphically, this implies that the parabola represented by the quadratic equation just "touches" the x-axis at the double root without crossing it. The condition for a double root in a standard quadratic equation \(y = x^2 + bx + c\) is \(b^2 = 4c\). This easy-to-check condition simplifies the process of identifying double roots without solving the equation explicitly.

Roots of Equations

The roots of a quadratic equation are essentially the values of \(x\) where the equation equals zero. These roots are the solutions we find when solving the equation \(ax^2 + bx + c = 0\). In essence, they are the x-coordinates where the corresponding parabola intersects the x-axis.

For a general quadratic equation, we apply the quadratic formula to find the roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The "\(\pm\)" indicates that there can be two solutions, corresponding to the intersections mentioned.

The nature of these roots—whether they're distinct, double, or complex—is determined by the value of the discriminant \(\Delta\):

• Two distinct real roots when \(\Delta > 0\).
• Exactly one (double) real root when \(\Delta = 0\).
• Two complex roots when \(\Delta < 0\).

By analyzing the discriminant before solving with the quadratic formula, we can anticipate the nature of the roots and approach the solution process with deeper insight.