Problem 86 - 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. $$8 x^{2}=-14 x-3$$

Step-by-Step Solution

Answer

The equation has two distinct rational solutions.

Step 1: Rewrite the equation in standard form

Rewrite the given quadratic equation in the standard form of \[ ax^2 + bx + c = 0 \]. The given equation is \[ 8x^2 = -14x - 3 \]. To get this into standard form, add \(14x\) and \(3\) to both sides: \[ 8x^2 + 14x + 3 = 0 \]

Step 2: Identify coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the standard form equation \[ 8x^2 + 14x + 3 = 0 \]string: \[ a = 8, \ b = 14, \ c = 3 \]

Step 3: Calculate the discriminant

Use the formula for the discriminant \[ \text{Discriminant} = b^2 - 4ac \] Substitute the values of \(a\), \(b\), and \(c\): \[ \text{Discriminant} = 14^2 - 4(8)(3) = 196 - 96 = 100 \]

Step 4: Interpret the discriminant

Use the value of the discriminant to determine the nature of the roots: - If the discriminant is positive and a perfect square (like 100), the equation has two distinct rational solutions. - If the discriminant is positive but not a perfect square, the equation has two distinct irrational solutions. - If the discriminant is zero, the equation has exactly one rational solution. - If the discriminant is negative, the equation has two non-real complex solutions. Here, the discriminant is 100 which is a positive perfect square, so the equation has two distinct rational solutions.

Key Concepts

quadratic equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). These equations represent parabolas when plotted on a graph. The constants \(a, b,\text { and } c\) are coefficients where \(a eq 0\). Solving a quadratic equation means finding the values of \(x\) that make the equation true. Quadratic equations have different types of solutions depending on their discriminant.

nature of roots

The nature of roots of a quadratic equation can be determined by evaluating the discriminant. The discriminant is given by the formula: \[ \text{Discriminant} = b^2 - 4ac \]
The value of the discriminant tells us about the number and type of solutions:

• If the discriminant is positive and a perfect square, there are two distinct rational solutions.
• If the discriminant is positive but not a perfect square, there are two distinct irrational solutions.
• If the discriminant is zero, there is exactly one rational solution.
• If the discriminant is negative, the equation has two non-real complex solutions.

By calculating the discriminant, students can predict the kinds of solutions without solving the equation.

rational and irrational solutions

When solving quadratic equations, the solutions can be rational or irrational:

• Rational solutions* are numbers that can be expressed as a fraction of two integers. For example, \( \frac {3}{2} \) is rational.
• Irrational solutions are numbers that cannot be written as a simple fraction. Their decimal form is non-repeating and non-terminating, such as \( \root 2\root \).

Knowing the nature of the roots can help in understanding the solutions and the behavior of the quadratic function better.