Problem 11 - 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. $$ 3 x^{2}-5 x+2=0 $$

Step-by-Step Solution

Answer

The roots are \(x = 1\) and \(x = \frac{2}{3}\).

Step 1: Identify the coefficients

The quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Comparing with the given equation \(3x^2 - 5x + 2 = 0\), we identify the coefficients: \(a = 3\), \(b = -5\), and \(c = 2\).

Step 2: Write down the quadratic formula

The quadratic formula to find the roots of a quadratic equation \(ax^2 + bx + c = 0\) is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the discriminant

The discriminant \(D\) is given by \(b^2 - 4ac\). Substituting the values for \(a\), \(b\), and \(c\), we have:\[D = (-5)^2 - 4 \cdot 3 \cdot 2 = 25 - 24 = 1.\]

Step 4: Apply the quadratic formula using the discriminant

Since the discriminant is \(1\), which is positive, we will have two real roots. Using the quadratic formula:\[x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 3} = \frac{5 \pm 1}{6}.\]So the two roots are:\[x = \frac{5 + 1}{6} = \frac{6}{6} = 1\]and\[x = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}.\]

Key Concepts

Discriminant

When working with a quadratic equation, one key element to examine is the **discriminant**. The discriminant is found within the quadratic formula itself and is represented by the expression under the square root sign:

• It is denoted by the expression \(b^2 - 4ac\).

The primary role of the discriminant is to determine the nature of the roots of the quadratic equation. The value of the discriminant gives us insight into what kind of solutions we can expect:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, or in other words, a repeated real root.
• If \(D < 0\), the equation has no real roots, but two complex conjugate roots.

In our exercise, calculating \(D = 1\) means that we have two different real roots, as the discriminant is positive. This information greatly informs us on how to proceed when finding the roots.

Quadratic Equation

A **quadratic equation** is one of the most common algebraic equations you will encounter, featuring a variable raised to the second power.

• Its general form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

Our goal when solving a quadratic equation is to find the values of \(x\) (the roots) that make the equation true.
The quadratic formula is a reliable method for solving any quadratic equation:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Using the quadratic formula involves plugging the given coefficient values into the formula itself, such as what was done in the original exercise where \(a = 3\), \(b = -5\), and \(c = 2\). This step leads to a straightforward calculation route to find the roots.
Once you identify these coefficients, the rest is arithmetic and algebraic manipulation.

Irrational Roots

**Irrational roots** are a specific type of solution you can encounter while solving quadratic equations. They occur when the discriminant is a positive non-perfect square, causing the square root part of the quadratic formula to be an irrational number.
An irrational number cannot be expressed as a simple fraction or ratio of two integers, and therefore, it often requires a simplest radical form.

• In simplest radical form, the square root is left in its radical form, instead of being turned into a long decimal.
• For example, \(\sqrt{2}\) is the simplest radical form of an irrational number.

In our exercise, after calculating the discriminant, we found that it was a perfect square, specifically \(1\), leading to rational roots.
If the discriminant weren’t a perfect square, such as \(5\), we would end up with roots like \(\frac{-b \pm \sqrt{5}}{2a}\), which are irrational and can be expressed in simplest radical form for clarity.