Problem 35 - College Algebra

Question

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ 9 x^{2}-30 x+25=0 $$

Step-by-Step Solution

Answer

The discriminant is 0; there is one real and equal solution.

Step 1: Identify Coefficients

For the quadratic equation in the form \( ax^2 + bx + c = 0 \), the given equation is \( 9x^2 - 30x + 25 = 0 \). Here, \( a = 9 \), \( b = -30 \), and \( c = 25 \).

Step 2: Calculate the Discriminant

The discriminant, \( D \), for a quadratic equation is given by the formula \( D = b^2 - 4ac \). Substituting the values we have: \( b = -30 \), \( a = 9 \), and \( c = 25 \). Thus,\( D = (-30)^2 - 4(9)(25) = 900 - 900 = 0 \).

Step 3: Determine the Number and Nature of Solutions

If the discriminant \( D = 0 \), the quadratic equation has exactly one real solution. This means the roots are real and equal.

Key Concepts

Understanding the Discriminant

The discriminant is an important part of quadratic equations. It provides key insights into the nature of the solutions of the equation without actually solving it. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula \( D = b^2 - 4ac \). This is derived from the components of a quadratic equation itself.

Here’s what the discriminant tells you:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution — the roots are real and equal.
• If \( D < 0 \), there are no real solutions; the solutions are complex or imaginary.

This simple computation can save time, as it informs you of the nature of the solutions before attempting to find them.

Nature of Real Solutions

The real solutions to a quadratic equation depend directly on the discriminant value. When we say solutions are 'real', this means they exist on the number line and can be practically interpreted and used. For a quadratic equation based on its discriminant:

• When \( D > 0 \), the solutions are two different points on the number line — this describes two unique intersections with the x-axis when plotted on a graph.
• When \( D = 0 \), the solution is one point — a "double root" or more formally, a repeated solution. This happens because the quadratic "touches" the x-axis at just one point.
• For \( D < 0 \), potential solutions are not real and can't be placed on the number line; thus they manifest as complex numbers.

Understanding these scenarios helps in grasping how the quadratic relates to its graph and its interactions with axes.

Using the Quadratic Formula

The quadratic formula is a powerful equation-dealing tool that guarantees finding the roots of any quadratic equation \( ax^2 + bx + c = 0 \). It is written as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this formula, \( \pm \) indicates that there are potentially two solutions. This aspect introduces the role of the discriminant \( b^2 - 4ac \) under the square root sign. The nature and number of solutions are exactly as predicted by the discriminant. The quadratic formula turns computation into visualization:

• If \( b^2 - 4ac \) is positive, \( \sqrt{b^2 - 4ac} \) is real and yields two different results, hence producing two distinct solutions.
• If \( b^2 - 4ac \) is zero, the square root term vanishes, simplifying to one unique solution.
• When it's negative, the square root involves an imaginary number, leading to complex solutions.

Thus, solving a quadratic equation using this formula gives not just the solutions, but a visual insight into the characteristics of the quadratic function itself.