Problem 48 - Algebra for College Students

Question

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$-3 x^{2}+7 x=0$$

Step-by-Step Solution

Answer

The discriminant is 49. There are two distinct real solutions.

Step 1: Identify the equation format

Rewrite the given equation in the standard quadratic form: the standard equation is: a x^{2} + b x + c = 0 The given equation is: -3 x^{2} + 7 x = 0 a = -3, b = 7, c = 0

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation is given by: b^{2} - 4ac Substitute the values: b = 7, a = -3, and c = 0 b^{2} - 4(-3)(0) = 49 - 0 = 49

Step 3: Analyze the Discriminant

Now, determine the number of real solutions based on the value of the discriminant: If the discriminant is greater than 0, then there are two distinct real solutions. If the discriminant is equal to 0, then there is exactly one real solution. If the discriminant is less than 0, there are no real solutions. Since 49 is greater than 0, there are two distinct real solutions.

Key Concepts

quadratic equations

A quadratic equation is an equation of the form: \[a x^{2} + b x + c = 0\]. Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). This form is known as the standard form of a quadratic equation. Quadratic equations are very important in mathematics because they describe the parabolic relationship between variables.
You often find them in problems involving projectile motion, areas, and optimization. In the equation \(-3x^{2} + 7x = 0\), you can identify the coefficients: \(a = -3\), \(b = 7\), and \(c = 0\). This step is crucial as these coefficients are used in further calculations, especially when finding the discriminant, which helps determine the number of real solutions.

real solutions

Real solutions to quadratic equations are the points where the parabola (graph of the equation) crosses the x-axis. These solutions are the values of \(x\) that make the quadratic equation true. Depending on the discriminant, the quadratic equation can have:

• Two distinct real solutions
• One real solution
• No real solutions (when solutions are complex or imaginary)

When solving \(-3x^{2} + 7x = 0\), after identifying the coefficients and calculating the discriminant, you see that it determines how many real solutions the equation has. If the graph of the equation touches or intersects the x-axis, those points are your real solutions.

discriminant calculation

The discriminant is a key part of solving quadratic equations and is given by the formula \(b^{2} - 4ac\). This value tells you the nature and number of solutions for the quadratic equation.

• If \(b^{2} - 4ac > 0\), there are two distinct real solutions.
• If \(b^{2} - 4ac = 0\), there is exactly one real solution.
• If \(b^{2} - 4ac < 0\), there are no real solutions (solutions are complex).

For the equation \(-3 x^{2} + 7 x = 0\):
Given \(a = -3\), \(b = 7\), and \(c = 0\).
The discriminant, \(b^{2} - 4ac\), is calculated as follows: \[7^{2} - 4(-3)(0) = 49 - 0 = 49\]. Since the discriminant is 49, which is greater than 0, this means that the quadratic equation has two distinct real solutions. This analysis helps understand why and how many solutions exist, providing insight into the behavior of the parabolic graph.