Problem 8 - 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. $$ x^{2}-8=0 $$

Step-by-Step Solution

Answer

The roots of the equation are \( x = 2\sqrt{2} \) and \( x = -2\sqrt{2} \).

Step 1: Identify Coefficients

In the quadratic equation, compare with the standard form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 0 \), and \( c = -8 \).

Step 2: Apply the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 1 \), \( b = 0 \), and \( c = -8 \) into the formula.

Step 3: Calculate the Discriminant

The discriminant \( b^2 - 4ac \) must be calculated. Here, \( b^2 - 4ac = 0^2 - 4 \times 1 \times (-8) = 32 \).

Step 4: Simplify the Square Root

The square root of the discriminant \( \sqrt{32} \) can be expressed as \( \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \).

Step 5: Find the Roots

Substitute the calculated values back into the quadratic formula: \( x = \frac{-0 \pm 4\sqrt{2}}{2 \times 1} = \frac{\pm 4\sqrt{2}}{2} \). Simplify to find \( x = \pm 2\sqrt{2} \).

Key Concepts

Discriminant

The discriminant is a key part of the quadratic formula. It helps determine the nature of the roots of a quadratic equation. When we look at the quadratic formula, which is \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]the part under the square root sign, \( b^2 - 4ac \), is called the discriminant.

• If the discriminant is positive, the quadratic equation has two real and distinct roots.
• If it equals zero, there’s one real root, which is also known as a repeated or double root.
• If the discriminant is negative, there are no real roots, the roots are complex instead.

In this exercise, we calculated the discriminant as \( 32 \). Since \( 32 \) is positive, we have two real roots. They happen to be irrational because the square root of \( 32 \) doesn’t simplify to a whole number.

Simplest Radical Form

The simplest radical form is the most reduced version of a radical expression. When simplifying radicals, we look for perfect squares within the number under the root to help simplify it further. It's a way to express a square root that is as simplified as possible:

• First, identify any perfect square factors of the number.
• Write the square root of these factors as a product.
• Take the square root of the perfect square factor out of the radical.

In this problem, the discriminant was \( 32 \). We simplified \( \sqrt{32} \) by rewriting it as \( \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \). Therefore, the simplest radical form of \( \sqrt{32} \) is \( 4\sqrt{2} \). Simplifying the radicals efficiently leads to easier calculations and a more precise final answer.

Quadratic Equation Coefficients

In any quadratic equation, identifying the coefficients is the first step to solving it using the quadratic formula. The standard form of a quadratic equation is:\[ ax^2 + bx + c = 0 \]where:

• \( a \) is the quadratic coefficient
• \( b \) is the linear coefficient
• \( c \) is the constant term

For the equation \( x^{2} - 8 = 0 \), we compare it to the standard form to find:

• \( a = 1 \)
• \( b = 0 \)
• \( c = -8 \)

Understanding these coefficients is crucial, as they’re plugged into the quadratic formula to work out the solution. These numbers control the shape and position of the parabola that represents the quadratic equation, and by substituting them correctly into the formula, we can find the roots of the equation.