Problem 84 - 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. $$x^{2}+4 x+4=0$$

Step-by-Step Solution

Answer

The equation has one rational solution because the discriminant is 0.

Step 1: Identify coefficients

For the given quadratic equation \[x^2 + 4x + 4 = 0\], identify the coefficients as follows: - \(a = 1\) (coefficient of \(x^2\))- \(b = 4\) (coefficient of \(x\))- \(c = 4\) (constant term)

Step 2: Write the discriminant formula

The discriminant \(D\) of a quadratic equation \[ax^2 + bx + c = 0\] is given by the formula: \[D = b^2 - 4ac\]

Step 3: Substitute coefficients into the discriminant formula

Substitute the identified coefficients \(a = 1\), \(b = 4\), and \(c = 4\) into the discriminant formula: \[D = 4^2 - 4 \times 1 \times 4 = 16 - 16 = 0\]

Step 4: Analyze the discriminant

Since the discriminant \(D = 0\): - The equation has exactly one distinct real solution (repeated root).- The solution is rational because it is a perfect square.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree 2. It is generally represented in the form:
\[ax^2 + bx + c = 0\]
The variables in this equation are:

• \(a\)the coefficient of \(x^2\),
\(b\)the coefficient of \(x\), \(c\)the constant term,
• = 0 the quadratic equation must be equal to zero

These components form a U-shaped graph called a parabola when plotted.
By using specific methods, we can find the values of x that make the equation true.
These values are known as the solutions or roots of the quadratic equation.

Discriminant

The discriminant is a key part of the quadratic formula. It's found inside the square root in the quadratic formula and is used to determine the nature and number of the solutions of a quadratic equation.
The discriminant \(D\) for the equation \[ax^2 + bx + c = 0 \] is given by:
\[ D = b^2 - 4ac \]
Here’s why the discriminant is important:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has exactly one real solution (repeated root).
• If \(D < 0\), the equation has two complex (non-real) solutions.

Calculating the discriminant helps you predict these outcomes without having to solve the quadratic equation directly.

Real Solutions

Real solutions refer to the values of \(x\) that make the quadratic equation true and lie on the real number line.
When examining the discriminant:

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

As in our example \[x^2 + 4x + 4 = 0\], we found that \(D = 0\). This means there is one real solution.

Rational Roots

Rational roots are solutions to a quadratic equation that can be expressed as a simple fraction (\(p/q\) where \(p\) and \(q\) are integers) or whole numbers.
Roots can be classified concerning the type of numbers they belong to:

• A rational root is when the solution is an integer or a fraction.
• An irrational root is when the solution is not exactly expressible as a fraction; these roots often involve square roots of non-perfect squares.

Since the discriminant in the given example, \[x^2 + 4x + 4 = 0\], is zero (i.e., \[D = 0\]), the equation has one real solution: \( x = -2 \), which is a rational root because \( -2 \) is an integer.