Problem 83 - 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. $$x^{2}-8 x+16=0$$

Step-by-Step Solution

Answer

One distinct rational solution

Step 1 - Identify coefficients

In the quadratic equation \(x^2 - 8x + 16 = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = -8\), and \(c = 16\).

Step 2 - Write the discriminant formula

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

Step 3 - Substitute the coefficients into the formula

Substitute \(a = 1\), \(b = -8\), and \(c = 16\) into the discriminant formula. \[D = (-8)^2 - 4 \cdot 1 \cdot 16\].

Step 4 - Calculate the discriminant

Evaluate the expression: \[D = 64 - 64 = 0\].

Step 5 - Determine the nature of the roots

If \(D = 0\), the quadratic equation has exactly one distinct real solution (a repeated root). Since the discriminant is zero, this solution is also rational.

Key Concepts

quadratic equation

A quadratic equation is a second-degree polynomial equation in a single variable. The standard form is \[ax^2 + bx + c = 0\]. It is called 'quadratic' because 'quad' means square, which refers to the squared term \(x^2\). The coefficients are \(a\) for the squared term, \(b\) for the linear term, and \(c\) for the constant term.

• If \(a\), \(b\), and \(c\) are known, we can use them to analyze the equation.
• The solutions to this equation are called the 'roots' of the equation.

Knowing how to work with quadratic equations is essential because they frequently appear in mathematics and science problems.
We can solve quadratic equations using various methods, such as factoring, completing the square, or using the quadratic formula.

nature of roots

The nature of the roots of a quadratic equation depends on the discriminant. The discriminant \(D\) is given by \[D = b^2 - 4ac\]. It tells us about the number and type of roots without actually solving the equation.

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), it has exactly one distinct real solution (a repeated root).
• If \(D < 0\), there are no real solutions, but two distinct non-real complex solutions.

Understanding the nature of the roots helps in predicting whether the solutions are real or complex and whether they are repeated or distinct.

rational solutions

When we talk about rational solutions of a quadratic equation, we mean the solutions that can be expressed as a ratio of two integers. For a quadratic equation to have rational solutions, its discriminant \(D\) must be a perfect square (including zero).

• If \(D = 0\), the equation has one repeated rational root.
• If \(D\) is a perfect square and positive, the equation has two distinct rational roots.

In the example equation \(x^2 - 8x + 16 = 0\), the discriminant is zero \((D = 0)\), indicating one repeated rational root. Hence, the solutions are not only real but also rational.