Problem 12 - Elementary Algebra

Question

Calculate the discriminant and use it to determine the number and type of solutions. Do not solve. $$ -x 2-4 x+2=0 $$

Step-by-Step Solution

Answer

The equation has two distinct real and irrational solutions.

Step 1: Write the quadratic equation in standard form

The given equation is \(-x^2-4x+2=0\). To match it with the standard quadratic form \(ax^2 + bx + c = 0\), simply note that \(a = -1\), \(b = -4\), and \(c = 2\).

Step 2: Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula \(D = b^2 - 4ac\). Substitute \(a = -1\), \(b = -4\), and \(c = 2\) into this formula to get: \[D = (-4)^2 - 4(-1)(2)\] Simplify the expression: \(16 + 8 = 24\). Therefore, the discriminant is 24.

Step 3: Determine the number and type of solutions

The discriminant value tells us about the nature of the roots of the quadratic equation. Since the discriminant \(D = 24\) is positive and not a perfect square, it indicates there are two distinct real and irrational solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. It’s called "quadratic" because the highest power of the variable is squared, which in math terms is the exponent of 2.
Quadratic equations often appear in the format:

• ax^2 + bx + c = 0

Here, the letters:

• a, b, and c are known constants
• x represents the variable

In a quadratic equation, the constant 'a' cannot be zero because then the equation becomes linear. Quadratic equations are important as they model many natural phenomena and processes in mathematics and science.

Nature of Roots

The nature of the roots of a quadratic equation means the type of solutions the equation has. Using mathematical language, these roots are the values that satisfy the equation where it equals zero. The key to understanding the nature of these roots is the discriminant, which gives insight into the number and type of roots for any given quadratic equation.
Think of it like this:

• If the discriminant is positive, the quadratic equation has two real roots.
• If it's a perfect square, those two roots will be rational.
• If it’s not a perfect square, the roots are irrational.
• If the discriminant is zero, there is exactly one real root, also known as a repeated root.
• If the discriminant is negative, the roots are complex or imaginary—they do not have real number solutions.

Standard Form

The standard form of a quadratic equation is its typical structure, which is: ax^2 + bx + c = 0. Having the equation in this form allows for easier manipulation and calculation, especially when using standard formulas like the one for computing the discriminant.
To align an equation with the standard form, you may need to rearrange the terms or change signs. For instance, converting

• -x^2 - 4x + 2 = 0

To match with its standard form labels them as:

• a = -1
• b = -4
• c = 2

Real and Irrational Solutions

Real and irrational solutions refer to the types of numbers the solutions to a quadratic equation can be. Solutions of a quadratic equation appear where its graph intersects the x-axis.

• Real solutions are those that you can locate on the number line, unlike imaginary solutions which are not. Real solutions can be either rational or irrational.
• Irrational solutions specifically are numbers that cannot be expressed as a simple fraction, meaning they have non-repeating, infinite decimal expansions. Examples include \(\sqrt{2}\) and \(\sqrt{3}\).
• The presence of irrational solutions is often signaled by a positive, non-perfect square discriminant in the context of quadratic equations.
In practice, when the discriminant of a quadratic equation is positive and not a perfect square, as in the example with
• D = 24,
the solutions will be two distinct real and irrational numbers.