Problem 88 - Precalculus Student Solutions Manual 5th

Question

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$2 x^{2}+4 x+1=0$$

Step-by-Step Solution

Answer

The equation has two distinct irrational solutions.

Step 1: Identify coefficients

Identify the coefficients from the quadratic equation. For the equation 2x^2 + 4x + 1 = 0: a = 2, b = 4, c = 1.

Step 2: Calculate the discriminant

Use the formula for the discriminant, D = b^2 - 4ac. Substitute the values for a, b, and c: D = 4^2 - 4 * 2 * 1.

Step 3: Simplify the discriminant

Simplify the equation to find the value of the discriminant: D = 16 - 8 = 8.

Step 4: Analyze the discriminant

Evaluate the discriminant to predict the number and type of solutions: Since D = 8 > 0, the quadratic equation has two distinct real solutions. Moreover, since the discriminant is not a perfect square, the solutions are irrational.

Key Concepts

quadratic equations

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This type of equation can describe a parabola in a graph and is fundamental in algebra. Quadratic equations can have different types of solutions, which we can predict using the discriminant. Let's explore how to do this in the context of the provided example: \(2x^2 + 4x + 1 = 0\).

discriminant

In the context of quadratic equations, the discriminant is a value that helps to determine the nature of the roots. It's calculated using the formula: \(D = b^2 - 4ac\). For the given equation (\(2x^2 + 4x + 1 = 0\)), we identify the coefficients as follows:

• \(a = 2\)
• \(b = 4\)
• \(c = 1\)

By substituting these into the discriminant formula, we calculate: \(D = 4^2 - 4(2)(1)\), leading to: \(D = 16 - 8 = 8\). This value tells us important information about the nature of the solutions.

real solutions

Real solutions are the values for \(x\) that satisfy the quadratic equation and are real numbers, as opposed to imaginary or complex numbers. To determine the type of solutions, we can analyze the discriminant (\(D\)):

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is one real solution (a repeated root).
• If \(D < 0\), there are no real solutions; instead, there are two complex solutions.

For \(2x^2 + 4x + 1 = 0\), since \(D = 8 > 0\), it has two distinct real solutions.

irrational solutions

Irrational solutions are real numbers that cannot be expressed as a simple fraction. They are typically expressed in terms of square roots. When the discriminant is a positive number that is not a perfect square (like 8 in our example), the roots of the quadratic equation are irrational. For \(2x^2 + 4x + 1 = 0\), since \(D = 8\) is positive but not a perfect square, the equation has two distinct irrational solutions. Understanding these aspects helps us predict the nature of the solutions without actually solving the equation.