Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, 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; one real solution exists.

Step 1: Identify the coefficients

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

Step 2: Write the discriminant formula

The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\).

Step 3: Substitute the coefficients into the discriminant formula

Substitute \(a = 9\), \(b = -30\), and \(c = 25\) into the formula: \[ D = (-30)^2 - 4 \times 9 \times 25 \]

Step 4: Calculate the discriminant

Calculate the value of the discriminant: \((-30)^2 = 900\) and \(4 \times 9 \times 25 = 900\), so \[D = 900 - 900 = 0\]

Step 5: Determine the number and nature of the solutions

If the discriminant \(D > 0\), there are two distinct real solutions. If \(D = 0\), there is exactly one real solution (a repeated root). If \(D < 0\), there are two complex conjugate solutions. Here, since \(D = 0\), the equation has one real solution.

Key Concepts

Understanding the Nature of Solutions

When dealing with quadratic equations, the nature of the solutions is primarily determined by the discriminant, denoted as \(D\). The discriminant is calculated from the coefficients of the quadratic equation \(ax^2 + bx + c = 0\) using the formula \(D = b^2 - 4ac\). The value of \(D\) indicates the type of solutions we can expect:

• If \(D > 0\), the equation has two distinct real solutions. This usually means that the graph of the quadratic function crosses the x-axis at two different points.
• If \(D = 0\), the equation has exactly one real solution, also known as a repeated or double root. Here, the graph touches the x-axis at a single point.
• If \(D < 0\), the solutions are two complex conjugate numbers, implying that the graph does not intersect the x-axis at all.

In our example, since \(D = 0\), the quadratic equation has one real solution.

The Quadratic Formula in Detail

The quadratic formula is a reliable method for finding the solutions of a quadratic equation. An equation in the form \(ax^2 + bx + c = 0\) can be solved using this formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula not only provides the solutions but also incorporates the discriminant \(b^2 - 4ac\), making it versatile in determining the type and nature of these solutions. Depending on the value under the square root (the discriminant), this formula yields:

• Two distinct real solutions if the discriminant is positive.
• One real repeated solution if the discriminant is zero.
• Two complex solutions when the discriminant is negative. In such cases, \(\pm \sqrt{b^2 - 4ac}\) will involve imaginary numbers.

Using the quadratic formula allows you to derive solutions systematically without graphing the equation, which is especially useful for complex or unsolvable by factorization quadratics.

Determining the Number of Solutions

The discriminant, calculated from the quadratic equation \(ax^2 + bx + c = 0\) with \(D = b^2 - 4ac\), directly influences the number of solutions of the equation:

• When \(D > 0\), the equation has two solutions, indicating two x-intercepts if plotted on a graph. These solutions are distinct and real.
• When \(D = 0\), the equation simplifies to having just one solution, showing a single x-intercept where the parabola just touches the x-axis.
• If \(D < 0\), the equation yields no real solutions, offering two complex solutions instead, which do not cross the x-axis.

Thus, the discriminant serves as a vital tool in quickly assessing how many solutions exist, providing a clear preview of what to expect before anyone attempts solving the equation.