Problem 41 - College Algebra

Question

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 3 x^{2}-5 x+1=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5 + \sqrt{13}}{6}\) and \(x = \frac{5 - \sqrt{13}}{6}\).

Step 1: Identify the Coefficients

In the quadratic equation \(3x^2 - 5x + 1 = 0\), identify the coefficients: \(a = 3\), \(b = -5\), and \(c = 1\).

Step 2: Write the Quadratic Formula

The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Compute the discriminant \(b^2 - 4ac\), where \(b = -5\), \(a = 3\), and \(c = 1\). \[(-5)^2 - 4 \times 3 \times 1 = 25 - 12 = 13\]

Step 4: Evaluate the Square Root

Calculate the square root of the discriminant: \(\sqrt{13}\). Since 13 is positive, proceed to calculate the roots.

Step 5: Calculate the Roots

Substitute the values back into the quadratic formula: \[x = \frac{-(-5) \pm \sqrt{13}}{2 \times 3}\]\[x = \frac{5 \pm \sqrt{13}}{6}\]

Step 6: Simplify the Expression

Thus, the solutions are: \[x_1 = \frac{5 + \sqrt{13}}{6}\]\[x_2 = \frac{5 - \sqrt{13}}{6}\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. It's a universal method applicable when other approaches like factoring are not possible or straightforward. Quadratic equations have the standard format: \[ ax^2 + bx + c = 0 \] The formula to find the solutions (or roots) of this equation is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here's how it works:

• a, b, and c are coefficients from the equation.
• \( \pm \) indicates that two solutions are being calculated - adding the square root for one and subtracting it for the other.
• \( \sqrt{b^2 - 4ac} \) represents the square root of the discriminant.

When using the quadratic formula, it's essential to identify these coefficients correctly for accurate solutions. Once you have them, you can substitute them into the formula and simplify to find the values of \( x \).

Discriminant

The discriminant is a crucial part of the quadratic formula that determines the nature of the roots of the equation. It is located under the square root in the formula: \[ b^2 - 4ac \] This term can reveal different types of solutions:

• **Positive Discriminant**: If \( b^2 - 4ac \) is positive, the quadratic equation has two distinct real roots. This indicates that the parabola cuts the x-axis at two different points.
• **Zero Discriminant**: When \( b^2 - 4ac \) is zero, there is exactly one real root. This means the parabola touches the x-axis at one point, known as a double root.
• **Negative Discriminant**: If \( b^2 - 4ac \) is negative, the equation has no real roots. Instead, the solutions are complex or imaginary, meaning the parabola does not intersect the x-axis.

Understanding the discriminant is essential, especially when determining whether a quadratic equation can be solved with real numbers or if it involves complex numbers.

Real and Imaginary Roots

Roots of a quadratic equation can either be real or imaginary, and recognizing their nature is integral to solving these equations effectively. Let's explore the concept further: - **Real Roots** are the solutions where the graph of the quadratic curve (a parabola) touches or crosses the x-axis. These occur when the discriminant \( b^2 - 4ac \) is zero or positive.
- **Imaginary Roots** arise when the discriminant \( b^2 - 4ac \) is negative. In such cases, the quadratic equation has solutions that cannot be expressed as real numbers. Instead, they involve complex numbers, which are in the form of \( a + bi \), where \( i \) is the square root of -1.
**Example:** In our original quadratic equation \( 3x^2 - 5x + 1 = 0 \):

• The calculated discriminant is 13, a positive number.
• This positivity of the discriminant tells us that the equation has two distinct real roots.

Understanding whether roots are real or imaginary helps visualize the solutions on a graph of the quadratic equation and indicates the intersection points with the x-axis.