Problem 77 - Algebra and Trigonometry

Question

Find all values of \(k\) that ensure that the given equation has exactly one solution. $$ 4 x^{2}+k x+25=0 $$

Step-by-Step Solution

Answer

The values of \( k \) are 20 and -20.

Step 1: Identify the Discriminant Formula

The discriminant of a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). For the quadratic equation to have exactly one solution (a repeated root), its discriminant must be zero.

Step 2: Set Up the Discriminant Equation

For the equation \( 4x^2 + kx + 25 = 0 \), identify \( a = 4 \), \( b = k \), and \( c = 25 \). The discriminant \( b^2 - 4ac \) becomes \( k^2 - 4(4)(25) \).

Step 3: Solve for the Discriminant Equal to Zero

Set the discriminant \( k^2 - 4(4)(25) = 0 \) equal to zero and solve for \( k \). This simplifies to \( k^2 - 400 = 0 \).

Step 4: Simplify the Equation

Factor the equation \( k^2 - 400 = 0 \) as \( (k - 20)(k + 20) = 0 \).

Step 5: Solve for Values of \( k \)

By solving \( (k - 20)(k + 20) = 0 \), we find \( k = 20 \) and \( k = -20 \). These are the values of \( k \) that ensure the quadratic equation has exactly one solution.

Key Concepts

Understanding the Discriminant

In the realm of quadratic equations, the discriminant plays a pivotal role in determining the nature of the roots. When dealing with a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is calculated using the formula \( b^2 - 4ac \). The discriminant gives us insight into how many roots the quadratic equation will have, and their nature—whether they are real or complex.

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), the equation has exactly one root, meaning the roots are real and identical (a repeated root).
• If \( b^2 - 4ac < 0 \), the equation has two complex roots.

The discriminant is particularly helpful because it avoids directly solving the equation to understand its properties. For the exercise question, when we compute \( k^2 - 400 \), setting it to zero will help us find values of \( k \) that make the quadratic equation have exactly one solution.

Exploring Repeated Roots

Repeated roots, or sometimes referred to as double roots, occur when a quadratic equation has a discriminant of zero. This means both roots of the equation are the same, and the parabola touches the x-axis at exactly one point.
To visualize this, imagine the parabola that describes a quadratic equation. Normally, it might intersect the x-axis twice, touch it once, or not at all when the roots are complex.
With repeated roots:

• The vertex of the parabola is exactly on the x-axis.
• The equation can be represented as \((x - r)^2 = 0\), where \(r\) is the repeated root.

In the given exercise, solving \( k^2 - 400 = 0 \) provides values of \( k \) where the quadratic has repeated roots, meaning a single solution at that point.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. Several methods can be employed to find the roots:

• Factorization: Expressing the quadratic as a product of two binomials.
• Completing the Square: Rewriting the equation in the form \( (x - p)^2 = q \).
• Quadratic Formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In the provided exercise, solving \( k^2 - 400 = 0 \) simplifies to solving \( (k - 20)(k + 20) = 0 \), a straightforward application of factorization.
This results in two possible solutions for \( k \), showing that these values of \( k \) make the equation have exactly one root, confirming the presence of repeated roots.Remember, understanding these methods enhances problem-solving flexibility and comprehension of quadratic behaviors.