Problem 69 - College Algebra Essentials

Question

Solve each equation in Exercises \(65-74\) using the quadratic formula. $$ 3 x^{2}-3 x-4=0 $$

Step-by-Step Solution

Answer

The roots of the given quadratic equation are \( x = \frac{3 + \sqrt{57}}{6} \) and \( x = \frac{3 - \sqrt{57}}{6} \).

Step 1: Compute discriminant

The first step is to compute the discriminant which is \( D = b^2 - 4ac \). Substituting the corresponding values \( a = 3 \), \( b = -3 \), and \( c = -4 \), we get \( D = (-3)^2 - 4(3)(-4) = 9 + 48= 57 \).

Step 2: Calculate the roots using Quadratic Formula

Next, use the quadratic formula to calculate the roots. The general form of the quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where plus-minus symbol signifies that there will be two solutions. Substitute \( D \), \( a \), \( b \) into the formula we get \( x = \frac{3 \pm \sqrt{57}}{6} \). So the roots of the equation are \( x = \frac{3 + \sqrt{57}}{6} \) and \( x = \frac{3 - \sqrt{57}}{6} \).

Key Concepts

Discriminant in Quadratic Equations

In quadratic equations, the discriminant plays a vital role. It helps determine the nature of the roots of the equation. The discriminant is represented by the symbol \( D \) and is calculated using the formula \( D = b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are the coefficients in the quadratic equation \( ax^2 + bx + c = 0 \). This formula is essential to understand before solving a quadratic equation with the quadratic formula.

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), the equation has two complex roots which are conjugates of each other.

Knowing the discriminant helps you predict the type of solutions before solving the equation fully, making it a useful checkpoint.

Solving Quadratic Equations

Solving quadratic equations is a common task in algebra that can reveal many mathematical relationships. One of the most reliable methods for solving them is the quadratic formula. This formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), allows you to find the roots of any quadratic equation with the structure \( ax^2 + bx + c = 0 \). It involves substituting the coefficients \( a \), \( b \), and \( c \) into the equation.
When applying this formula:

• Calculate the discriminant \( D = b^2 - 4ac \), which determines how the quadratic formula is applied.
• Substitute \( b \) and \( D \) into the formula.
• Solve for both \( x = \frac{-b + \sqrt{D}}{2a} \) and \( x = \frac{-b - \sqrt{D}}{2a} \) to find the two potential solutions for \( x \).

The quadratic formula is particularly useful as it provides exact solutions and applies universally to all quadratic equations, regardless of the nature of the coefficients.

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions that satisfy the equation. In mathematical terms, if you substitute a root back into the original equation, it should equal zero. These roots are derived from the quadratic formula or other solving methods such as factoring and completing the square. In the context of the quadratic formula, the roots are given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Let's break this down:

• When the discriminant \( D \) is positive, the square root in the formula provides two distinct real number results, leading to two different roots.
• When \( D = 0 \), the formula simplifies as the square root expression becomes zero, giving a single repeated root.
• For \( D < 0 \), the result involves the square root of a negative number, which introduces complex numbers in the roots.

Understanding the nature of roots can assist in knowing the behavior of the quadratic function graphically and numerically. Such insights can be key in applied fields like physics and engineering where understanding exact solutions is necessary.