Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+5 x-2=0 $$

Step-by-Step Solution

Answer

The roots are \(\frac{-5 + \sqrt{33}}{2}\) and \(\frac{-5 - \sqrt{33}}{2}\).

Step 1: Identify the coefficients

To use the quadratic formula, we need to identify the coefficients of the quadratic equation. From the equation \(x^2 + 5x - 2 = 0\), we have: \(a = 1\), \(b = 5\), and \(c = -2\).

Step 2: Write down the quadratic formula

The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]We will use this formula to solve for the roots of the equation.

Step 3: Calculate the discriminant

The discriminant is \(b^2 - 4ac\). Let's calculate this value using our coefficients:\[ b^2 - 4ac = 5^2 - 4 \times 1 \times (-2) = 25 + 8 = 33 \]Since the discriminant is positive and not a perfect square, we'll have two irrational roots.

Step 4: Substitute into the quadratic formula

Substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula:\[ x = \frac{-5 \pm \sqrt{33}}{2 \times 1} \]

Step 5: Simplify the expression

We simplify the expression for the two possible roots:- First root: \[ x = \frac{-5 + \sqrt{33}}{2} \]- Second root: \[ x = \frac{-5 - \sqrt{33}}{2} \]These are the simplest radical form for the irrational roots of the equation.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental part of algebra and mathematics in general. A quadratic equation is typically expressed in the form \( ax^2 + bx + c = 0 \), where:

• \( a \), \( b \), and \( c \) are constants
• \( a eq 0 \) (if \( a = 0 \), it becomes a linear equation, not quadratic)

The variable \( x \) represents the unknown or variable for which we seek solutions or "roots." These equations typically have two solutions, which are the values that satisfy the equation when substituted back into \( x \). Sometimes, these roots can be real (rational or irrational) or complex numbers. Quadratic equations can be solved by various methods, including factoring, completing the square, or using the quadratic formula.

The Role of the Discriminant

The discriminant is a crucial element in understanding the nature of the solutions of quadratic equations. It is derived from the coefficients of the quadratic equation \( ax^2+bx+c=0 \) and is calculated as:\[ b^2 - 4ac \]The value of the discriminant gives us insight into the type of roots the equation will have:

• If the discriminant is positive and a perfect square, the roots are real and rational.
• If the discriminant is positive but not a perfect square, the roots are real and irrational.
• If the discriminant is zero, there is exactly one real rational root (a repeated root).
• If the discriminant is negative, there are two complex conjugate roots.

In the context of the quadratic equation provided, the discriminant was calculated to be 33, which is positive and not a perfect square. This indicates that the equation has two distinct real irrational roots.

Exploring Irrational Roots

Irrational roots are types of roots that cannot be expressed as a simple fraction or integer. They often occur under a square root sign due to a discriminant that is positive but not a perfect square. When quadratic equations have irrational roots, they usually involve radicals. For instance, the roots of the equation\( x^2 + 5x - 2 = 0 \)were found using the quadratic formula with a discriminant of 33. The resulting roots are:

• \( \frac{-5 + \sqrt{33}}{2} \)
• \( \frac{-5 - \sqrt{33}}{2} \)

These expressions include \(\sqrt{33}\), which is an irrational number since 33 is not a perfect square. Irrational roots are useful in many applications, including engineering and physics, where precise calculations are necessary without rounding. Understanding how to handle these roots is vital for solving complex mathematical problems effectively.