Problem 69 - Intermediate Algebra

Question

Solve each of the following equations for \(x\). $$ x^{2}-5 a x+6 a^{2}=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3a\) and \(x = 2a\).

Step 1: Identify the Coefficients

The given quadratic equation is \(x^2 - 5ax + 6a^2 = 0\). We need to identify the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -5a\), and \(c = 6a^2\).

Step 2: Apply the Quadratic Formula

To solve the quadratic equation, we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the coefficients identified in Step 1 into this formula.

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\).\ The discriminant is \((-5a)^2 - 4(1)(6a^2) = 25a^2 - 24a^2 = a^2\).

Step 4: Solve for x

Use the quadratic formula with the calculated discriminant: \[x = \frac{-(-5a) \pm \sqrt{a^2}}{2(1)}\].\ Simplifying, we get \[x = \frac{5a \pm a}{2}\].\ There are two solutions: \[x_1 = \frac{5a + a}{2} = 3a\] and \[x_2 = \frac{5a - a}{2} = 2a\].

Key Concepts

Discriminant

The discriminant is a key part of the quadratic formula and acts as a critical tool in determining the nature of the roots of a quadratic equation. In the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), there is a term under the square root called the discriminant, represented by \( \Delta = b^2 - 4ac \).

• If the discriminant \( \Delta \) is greater than zero, the quadratic equation has two distinct real roots.
• If \( \Delta \) is equal to zero, there is one real root, also known as a repeated or double root.
• If \( \Delta \) is less than zero, the quadratic equation has no real roots; instead, it has two complex conjugates.

In our example, the discriminant is calculated as \( a^2 \), which is positive as long as \( a eq 0 \). This means our equation has two distinct real roots. The discriminant guides us in understanding how the curve of the quadratic equation behaves without necessarily computing the entire solution.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \( x \). The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \) because otherwise, it would not be a quadratic equation.
Quadratic equations can be solved by different methods such as:

• Factoring
• Completing the square
• Using the quadratic formula
• Graphical method

In this exercise, we solved the quadratic equation \( x^2 - 5ax + 6a^2 = 0 \) using the quadratic formula, which is a straightforward and reliable method that works for any quadratic equation. Understanding each component of the equation helps in visualizing the problem and selecting the most efficient solution method.

Coefficients

In a quadratic equation \( ax^2 + bx + c = 0 \), coefficients \( a \), \( b \), and \( c \) play an important role as they define the specific shape and position of the parabola represented by the equation.

• \( a \) is the coefficient of \( x^2 \) and it determines the direction of the parabola (upward if \( a > 0 \), downward if \( a < 0 \)) and its "width."
• \( b \) is the coefficient of \( x \) and affects the parabola's axis of symmetry and the position of the vertex without changing its direction.
• \( c \) is the constant term and determines the point where the parabola intersects the y-axis.

Identifying these coefficients correctly is crucial when applying the quadratic formula. In our equation, they are: \( a = 1 \), \( b = -5a \), and \( c = 6a^2 \). This step forms the foundation for solving the equation accurately using the quadratic formula.