Problem 32 - Basic Technical Mathematics with Calculus

Question

Solve the given applied problem. Find \(c\) such that \(y=x^{2}-12 x+c\) has exactly one real root.

Step-by-Step Solution

Answer

\(c = 36\) ensures \(y = x^2 - 12x + c\) has one real root.

Step 1: Understanding the Condition for One Real Root

A quadratic equation has exactly one real root when its discriminant is zero. The standard form of a quadratic equation is \(ax^2 + bx + c\), and the discriminant \(\Delta\) is given by \(\Delta = b^2 - 4ac\). For the equation to have one real root, we need \(\Delta = 0\).

Step 2: Identify Coefficients

In the equation \(y = x^2 - 12x + c\), we can identify the coefficients as \(a = 1\), \(b = -12\), and \(c = c\).

Step 3: Apply the Discriminant Formula

Substituting the coefficients into the discriminant formula, we need \((-12)^2 - 4(1)(c) = 0\). Simplifying gives \(144 - 4c = 0\).

Step 4: Solve for \(c\)

To find \(c\), solve the equation \(144 - 4c = 0\). Simplifying further, we rearrange to get \(4c = 144\). Divide both sides by 4 to get \(c = 36\).

Key Concepts

Discriminant

The term "discriminant" refers to a specific part of the quadratic formula that helps decide the nature of the roots of a quadratic equation. Here's a simple way to understand it:

• In a standard quadratic equation, expressed as \( ax^2 + bx + c \), the discriminant \( \Delta \) is calculated using the formula \( \Delta = b^2 - 4ac \).
• The discriminant tells us about the number and type of roots the quadratic equation will have.

When applying this to real life equations, the value of \( \Delta \) is crucial:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), it has exactly one real root (a repeated root).
• If \( \Delta < 0 \), there are no real roots, but two complex conjugate roots instead.

In the given problem, we needed the quadratic to have exactly one real root, which led us to set \( \Delta = 0 \). This balanced the equation so that both factors in the formula equalize, showing a perfect square root solution.

Real Roots

Roots of a quadratic equation are the solutions to the equation \( ax^2 + bx + c = 0 \). These roots could be real numbers or complex numbers, and understanding the nature of these roots is fundamental to solving quadratic equations.
The notion of 'exactly one real root' occurs in special cases when the discriminant \( \Delta \) is zero:

• Mathematically, this implies the equation has a double root or a repeated root at the same numerical value.
• This situation manifests at the vertex of the parabola described by the quadratic function.
• Graphically, the parabola just touches the x-axis at one point.

In essence, an equation with one real root is neatly "balanced," offering insight that the number and nature of roots intertwine closely with the discriminant's outcome.
For practical problems such as the one in our exercise, determining conditions for having exactly one real root is often central to ensuring solutions meet necessary criteria.

Solving Quadratics

Solving quadratic equations can be straightforward or require some calculations, depending on the specific form and details of the equation. Here, let's explore the most common methods:

• Factoring: When a quadratic expression can be neatly factored, this is often the fastest technique to find the roots.
• Completing the Square: This method finds the vertex form of a quadratic, isolating \( x \) by adjusting terms, and it works in any situation.
• Quadratic Formula: Solutions can always be found using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula incorporates the discriminant and is reliable when other methods are ineffective.

In our exercise, we used the discriminant component of the quadratic formula, \( b^2 - 4ac = 0 \) specifically, to deduce the condition for exactly one real root. Ultimately, understanding these problem-solving strategies paves the way for tackling diverse real-world and theoretical problems where quadratic equations come into play.