Problem 109 - Algebra for College Students

Question

Write a quadratic equation of each of the following types, then trade your equations with those of a classmate. Solve the equations and verify that they are of the required types. a) a single rational solution b) two rational solutions c) two irrational solutions d) two imaginary solutions

Step-by-Step Solution

Answer

a) \(x^2 - 2x + 1 = 0 \), b) \(x^2 - 5x + 6 = 0\), c) \(x^2 - 2x - 1 = 0\), d) \(x^2 + x + 1 = 0\).

Step 1: Identify type-specific quadratic equations

We need to write a quadratic equation for each specified type and then solve them. Quadratic equations have the form: \[ ax^2 + bx + c = 0 \]

Step 2: Single rational solution

For a single rational solution, the discriminant (\[ b^2 - 4ac \]) must be zero. An example equation is: \[ x^2 - 2x + 1 = 0 \]. Solve: \[ (x - 1)^2 = 0, \text{ Solution: } x = 1 \]

Step 3: Two rational solutions

For two rational solutions, the discriminant (\[ b^2 - 4ac \]) must be positive and a perfect square. An example equation is: \[ x^2 - 5x + 6 = 0 \]. Solve using factoring: \[ (x - 2)(x - 3) = 0, \text{ Solutions: } x = 2 \text{ or } x = 3 \]

Step 4: Two irrational solutions

For two irrational solutions, the discriminant (\[ b^2 - 4ac \]) must be positive and not a perfect square. An example equation is: \[ x^2 - 2x - 1 = 0 \]. Solve: \[ x = \frac{2 \pm \sqrt{4 + 4}}{2}, \text{ Solutions: } x = 1 \pm \sqrt{2} \]

Step 5: Two imaginary solutions

For two imaginary solutions, the discriminant (\[ b^2 - 4ac \]) must be negative. An example equation is: \[ x^2 + x + 1 = 0 \]. Solve using the quadratic formula: \[ x = \frac{-1 \pm \sqrt{1 - 4}}{2}, \text{ Solutions: } x = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i \]

Key Concepts

discriminant

The discriminant plays a crucial role in determining the nature of the solutions to a quadratic equation. For a quadratic equation in the form -3 + 6 = 0, the discriminant is given by The discriminant helps you understand the types of solutions without actually solving the equation. It is a quick way to predict whether you will get real or imaginary solutions.

rational solutions

Rational solutions are solutions to a quadratic equation that can be expressed as a ratio of two integers, such as 1/2, -3, or 4. When the discriminant ( b^2 - 4ac) is a non-negative perfect square, it means that the quadratic equation has rational solutions. For example, consider the equation x^2 - 5x + 6 = 0 is 1 (a perfect square).

irrational solutions

Irrational solutions occur when the discriminant ( b^2 - 4ac) is a positive number but not a perfect square. This results in solutions involving square roots of non-perfect squares. For example x^2 - 2x - 1 = 0 . The discriminant ( ) is positive but not rational solutions.

imaginary solutions

Imaginary solutions are formed when the discriminant ( b^2 - 4ac) is negative. This occurs because the square root of a negative number is not a real number, but an imaginary number. For example x^2 + x + 1 = 0, which results in imaginary solutions . This indicates the presence of complex numbers in the solutions sorted