Problem 27 - Contemporary Precalculus

Question

Find the number of real solutions of the equation by computing the discriminant. $$x^{2}+4 x+1=0$$

Step-by-Step Solution

Answer

Answer: The equation has two distinct real solutions.

Step 1: Identify the coefficients

The given equation is \(x^2 + 4x + 1 = 0\). Comparing it to the general form \(ax^2 + bx + c = 0\), we can identify the coefficients as follows: \(a = 1\), \(b = 4\), and \(c = 1\).

Step 2: Calculate the discriminant

We can compute the discriminant using the formula \(D = b^2 - 4ac\). In our case, it becomes: $$D = 4^2 - 4(1)(1) = 16 - 4 = 12$$.

Step 3: Determine the number of real solutions

Now that we have computed the discriminant, we can determine the number of real solutions. Since \(D = 12 > 0\), it means that the equation has two distinct real solutions.

Key Concepts

Discriminant

When solving quadratic equations, one important concept to understand is the discriminant. The discriminant helps us determine the nature of the roots of the quadratic equation. For any quadratic equation of the form:\( ax^2 + bx + c = 0 \)The formula to find the discriminant is:\( D = b^2 - 4ac \).

What does the discriminant tell us?

It's all about the value of \( D \):

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root, often called a repeated root.
• If \( D < 0 \), the equation has no real roots, but two complex roots instead.

By knowing the discriminant, you can predict whether the solutions are real or complex without actually solving the equation.In our example equation \( x^2 + 4x + 1 = 0 \), we calculated the discriminant as \( D = 12 \), indicating that it has two distinct real solutions.

Real Solutions

Real solutions indicate where the quadratic equation intersects the x-axis in a coordinate plane. These intersections (roots) are the solutions to the quadratic equation.Understanding whether we have real solutions or not is crucial for graphing and interpreting the behavior of quadratic functions.

Nature of real solutions

Real solutions can be either distinct or repeated:

• Distinct real solutions: Occur when the discriminant \( D > 0 \), meaning the parabola intersects the x-axis at two points.
• Repeated real solution: Occurs when \( D = 0 \), and the vertex of the parabola touches the x-axis at exactly one point.

In practical terms, real solutions are the x-values where the quadratic equation equals zero.In our example, since \( D = 12 > 0 \), we confirmed that the equation has two distinct real solutions. This means two separate x-values will solve \( x^2 + 4x + 1 = 0 \).

Coefficients

Coefficients are the numbers in front of the variables in a polynomial equation. In quadratic equations, they help us determine many critical attributes, like the discriminant.For the general quadratic equation:\( ax^2 + bx + c = 0 \)The coefficients are:

• \( a \): The coefficient of \( x^2 \)
• \( b \): The coefficient of \( x \)
• \( c \): The constant term

These coefficients are crucial because they help in calculating the discriminant using \( D = b^2 - 4ac \). In our example, the equation is \( x^2 + 4x + 1 = 0 \), where the coefficients are identified as:

• \( a = 1 \)
• \( b = 4 \)
• \( c = 1 \)

These play a fundamental role in further computations, such as finding the roots or graphing the parabola.