Problem 42 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate. $$ 3 x^{2}+6 x=1 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-3 + 2\sqrt{3}}{3} \) and \( x = \frac{-3 - 2\sqrt{3}}{3} \).

Step 1: Move the constant to one side

Start with the quadratic equation:\[ 3x^2 + 6x = 1 \]Subtract 1 from both sides to move the constant to one side:\[ 3x^2 + 6x - 1 = 0 \]

Step 2: Identify coefficients for the quadratic formula

The quadratic equation is in the form:\[ ax^2 + bx + c = 0 \]Identify the coefficients: \[ a = 3, \, b = 6, \, c = -1 \]

Step 3: Apply the quadratic formula

The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the coefficients into the formula:\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} \]

Step 4: Simplify the expression under the square root

Calculate the discriminant (the expression under the square root):\[ b^2 - 4ac = 6^2 - 4 \cdot 3 \cdot (-1) = 36 + 12 = 48 \]So the expression becomes:\[ x = \frac{-6 \pm \sqrt{48}}{6} \]

Step 5: Simplify the square root and solve

The sqrt(48) can be simplified as:\[ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \]Substitute this back into the quadratic formula expression:\[ x = \frac{-6 \pm 4\sqrt{3}}{6} \]Simplify by dividing the terms by the common factor 2:\[ x = \frac{-3 \pm 2\sqrt{3}}{3} \]

Step 6: Present the solutions

The solutions to the quadratic equation are:\[ x = \frac{-3 + 2\sqrt{3}}{3} \quad \text{and} \quad x = \frac{-3 - 2\sqrt{3}}{3} \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides solutions for the variable \( x \) by considering the coefficients \( a \), \( b \), and \( c \).
The formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula gives two solutions because of the \( \pm \) symbol, representing the two possible values x can take.

• The part under the square root, \( b^2 - 4ac \), is crucial for finding the nature of the solutions.
• Ensure to substitute all coefficients correctly to avoid calculation mistakes.

Simplifying calculations step-by-step helps in understanding the quadratic formula better.

Discriminant

The discriminant is the component \( b^2 - 4ac \) found under the square root in the quadratic formula. It plays a vital role in determining the nature of the roots of a quadratic equation. Here’s how:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If it is zero, \( b^2 - 4ac = 0 \), there is exactly one real solution, also known as a double root.
• If it is negative, \( b^2 - 4ac < 0 \), the equation has two complex roots, meaning no real solutions.

In our problem, the discriminant equaled 48, which is positive, indicating two distinct real solutions.
This understanding helps predict solution types before even applying the whole formula.

Simplification

Simplification is the process of reducing expressions to their simplest form, making them easier to interpret and solve. In this exercise, simplification occurs in multiple steps:

• Rewriting the equation in standard quadratic form \( ax^2 + bx + c = 0 \) makes it easier to identify coefficients.
• Calculating and simplifying the discriminant helps in figuring out the type of roots.
• Simplifying square roots, such as \( \sqrt{48} = 4\sqrt{3} \), reduces complexity.
• Finally, dividing by common factors in the solution simplifies further, making the solution cleaner: \( \frac{-3 \pm 2\sqrt{3}}{3} \).

A careful approach to simplification ensures accuracy and makes equations less cumbersome to solve.