Problem 57 - Intermediate Algebra

Question

For each quadratic equation, first use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$ 15 x^{2}+17 x-4=0 $$

Step-by-Step Solution

Answer

The equation has two real solutions: \( x = \frac{1}{5} \) and \( x = -\frac{4}{3} \).

Step 1: Understand the Quadratic Equation

The quadratic equation given is in the form \( ax^2 + bx + c = 0 \). Here, \( a = 15 \), \( b = 17 \), and \( c = -4 \).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( \Delta = b^2 - 4ac \). Substituting the values, we find \( \Delta = 17^2 - 4 \times 15 \times (-4) = 289 + 240 = 529 \).

Step 3: Analyze the Discriminant

The discriminant \( \Delta = 529 \) is a positive number. This means the quadratic equation has two real solutions.

Step 4: Solve the Quadratic Equation

Since the discriminant is positive, we use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions. Substituting the values, \( x = \frac{-17 \pm \sqrt{529}}{30} = \frac{-17 \pm 23}{30} \). This gives two solutions: \( x_1 = \frac{6}{30} = \frac{1}{5} \) and \( x_2 = \frac{-40}{30} = -\frac{4}{3} \).

Key Concepts

Discriminant

When dealing with quadratic equations, the discriminant is a valuable tool. It informs us about the nature of solutions for the equation. To put it simply, the discriminant is calculated using the formula \( \Delta = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). The discriminant helps to determine:

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, occurring twice, often called a repeated or equal root.
• If \( \Delta < 0 \), the solutions are nonreal complex numbers.

In our problem, the discriminant is 529. Since 529 is positive, this tells us that the quadratic equation has two distinct real solutions.

Real Solutions

Real solutions of a quadratic equation are the values of \( x \) that satisfy the equation and are real numbers. When we talk about real solutions, we're referring to solutions that can be plotted on a number line without involving imaginary numbers. These solutions come to life based on the discriminant:

• If the discriminant is positive, like in our current equation, we will have two real solutions, which means the graph of the equation will intersect the x-axis at two points.
• If the discriminant equals zero, the graph will touch the x-axis at just one point, indicating one real solution.

In the step-by-step solution, after identifying a positive discriminant, we confirmed that our equation indeed has two real solutions. This is a key element in solving quadratic equations, as it guides us to the right solving strategy.

Quadratic Formula

The quadratic formula is a universal tool for finding the solutions of any quadratic equation \( ax^2 + bx + c = 0 \). It is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this formula:

• \( \pm \) indicates that there will be two solutions: one with a plus and one with a minus.
• The part under the square root, i.e., \( \sqrt{b^2 - 4ac} \), is the discriminant, and its value determines whether the solutions are real or complex.

The quadratic formula allows us to find the exact points where the quadratic equation touches or crosses the x-axis. For our case:\[ x = \frac{-17 \pm \sqrt{529}}{30} \]This resolved to the real solutions \( \frac{1}{5} \) and \( -\frac{4}{3} \). This consistent formula provides a straightforward method to solve any quadratic equation, ensuring we can always find the solutions, provided we know the coefficients.