Problem 97 - Contemporary Precalculus

Question

Find a number \(k\) such that the given equation has exactly one real solution. $$k x^{2}+8 x+1=0$$

Step-by-Step Solution

Answer

Answer: \(k = 16\)

Step 1: 1. Identify the coefficients of the quadratic equation

In the given equation, \(kx^2+8x+1=0\), we can identify the coefficients as \(a=k\), \(b=8\), and \(c=1\).

Step 2: 2. Calculate the discriminant

The discriminant \(\Delta\) of the quadratic equation is given by the formula \(\Delta = b^2 - 4ac\). Substitute the coefficients we found in step 1, which are \(a=k\), \(b=8\), and \(c=1\), into the formula: $$\Delta = 8^2 - 4(k)(1) = 64 - 4k$$

Step 3: 3. Set the discriminant equal to zero

For the quadratic equation to have exactly one real root, the discriminant must be equal to zero. So, we have: $$64 - 4k = 0$$

Step 4: 4. Solve for \(k\)

Now, we need to solve the equation \(64 - 4k = 0\) to find the value of \(k\): $$4k = 64$$ Dividing both sides by 4, we get: $$k = 16$$ So, the value of \(k\) that makes the given quadratic equation \(kx^2 + 8x + 1=0\) have exactly one real solution is \(k = 16\).

Key Concepts

Understanding the Discriminant

The discriminant is a crucial part of solving quadratic equations because it helps determine the nature of the roots of the equation. It provides valuable insight into whether the solutions are real or complex and if the equation has one or more solutions.

In the context of a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using the formula \( \Delta = b^2 - 4ac \). This formula derives from the coefficients \( a \), \( b \), and \( c \) of the quadratic equation.

• When \( \Delta \) is greater than zero, the equation has two distinct real solutions.
• If \( \Delta \) is equal to zero, there is exactly one real solution, which means the roots are real and equal.
• When \( \Delta \) is less than zero, the solutions are complex or imaginary numbers.

This information simplifies understanding what kind of solutions you can expect. In our exercise, we set the discriminant to zero to find the specific value of \( k \) that ensures just one real solution exists. Calculating and understanding the discriminant ensures you can predict the types of solutions without directly solving the equation.

Exploring Real Solutions

A real solution in a quadratic equation occurs when the result of the equation lies on the real number line. Real solutions indicate that the parabola, which represents the quadratic equation, intersects the x-axis.

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant informs us about the nature of these intersections.

• Two distinct real solutions mean the parabola crosses the x-axis at two different points.
• Exactly one real solution implies the parabola touches the x-axis at one vertex point, known as a repeated root or a double root.

The process of determining whether solutions are real and how many there are is largely driven by the discriminant analysis. An important aspect of this is recognizing that when the discriminant equals zero, the vertex of the parabola is exactly on the x-axis, indicating one real solution. This understanding is especially useful in scenarios such as our problem where we need the equation to have precisely one real solution; the discriminant thus plays a key role.

Unpacking Quadratic Coefficients

Quadratic coefficients in an equation \( ax^2 + bx + c = 0 \) are the constants \( a \), \( b \), and \( c \), which dictate the equation's form and the curvature of its graph.

Each coefficient has a specific role:

• \( a \) is the quadratic coefficient, responsible for the parabola's "opening" direction and its width. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• \( b \) provides orientation with respect to the y-axis and impacts the axis of symmetry of the parabola.
• \( c \) is the constant term. It affects the vertical placement of the parabola on the graph.

Identifying these coefficients is the first step in solving a quadratic equation. In our exercise, understanding that \( a = k \), \( b = 8 \), and \( c = 1 \) allowed us to set up the equation for the discriminant. Without knowing these coefficients, solving for \( k \) to ensure one real solution would not be possible. This demonstrates their essential role in both forming and solving quadratic equations.