Problem 87 - 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. $$4 x^{2}=-6 x+3$$

Step-by-Step Solution

Answer

There are two distinct irrational solutions.

Step 1: Rewrite the equation in standard form

Rewrite the given equation in the form of a quadratic equation: \[4x^2 + 6x - 3 = 0\]

Step 2: Identify coefficients

Identify the coefficients from the standard form equation \[ax^2 + bx + c = 0\]Here, \(a = 4\), \(b = 6\), and \(c = -3\)

Step 3: Use the discriminant formula

The discriminant formula is given by \(\Delta = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(\Delta = 6^2 - 4(4)(-3)\)

Step 4: Calculate the discriminant

Calculate the discriminant: \(\Delta = 36 - 4(4)(-3) = 36 + 48 = 84\)

Step 5: Analyze the discriminant

Since the discriminant is positive (\(\Delta > 0\)), there are two distinct solutions. Because \(84\) is not a perfect square, the solutions are irrational. If the discriminant were zero, the equation would have one rational solution. And if it were negative, it would have two non-real complex solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of degree 2. It takes the standard form: \[ ax^2 + bx + c = 0 \] where:

• \(a\) is the coefficient of the squared term (\(x^2\)) and must be non-zero,
• \(b\) is the coefficient of the linear term (\(x\)), and
• \(c\) is the constant term.

The solutions to a quadratic equation are the values of \(x\) that satisfy the equation. These solutions can be found using various methods like factoring, completing the square, or the quadratic formula. The quadratic formula is given by: \[ x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } \] Where \( b^2 - 4ac \) is known as the discriminant. This formula provides a direct way to solve for the roots (solutions) of the quadratic equation.

Discriminant

The discriminant is an important part of the quadratic formula, given by: \[ \Delta = b^2 - 4ac \] The discriminant helps determine the nature and number of solutions for the quadratic equation. Here's how:

• If \( \Delta > 0 \), there are two distinct solutions.
• If \( \Delta = 0 \), there is exactly one solution (also known as a repeated or double root).
• If \( \Delta < 0 \), there are two complex solutions (which are non-real).

In the exercise, we used the discriminant to evaluate the quadratic equation: \[ 4x^2 + 6x - 3 = 0\] By identifying the coefficients \( a = 4 \), \( b = 6 \), and \( c = -3 \) and substituting into the discriminant formula, we found: \[ \Delta = 6^2 - 4(4)(-3) = 36 + 48 = 84 \] Since \( \Delta = 84 \) which is positive and not a perfect square, it indicates that the quadratic equation has two distinct irrational solutions.

Types of Solutions

Quadratic equations can have different types of solutions based on the value of the discriminant:

Real and Distinct Solutions

When \( \Delta > 0 \), the equation has two distinct real solutions. If \( \Delta \) is a perfect square (like 1, 4, 9), those solutions are rational. If not (like 2, 3, 84), they are irrational.

Real and Repeated Solutions

When \( \Delta = 0 \), the quadratic equation has exactly one solution, also known as a repeated or double root. This solution is always rational.

Complex Solutions

When \( \Delta < 0 \), the quadratic equation has two distinct complex solutions. These solutions are non-real and expressed in the form: \[ a \pm bi \] where \( i \) is the imaginary unit. Understanding these types of solutions can help you predict the nature of the roots without fully solving the equation. For the example in the exercise, our calculated discriminant of 84 indicates the quadratic equation has two distinct, irrational, real solutions.