Problem 36 - Algebra 1: Concepts and Skills

Question

Consider for quadratic equation \(y=2 x^{2}+6 x-3\). How many solutions does the equation have?

Step-by-Step Solution

Answer

The given quadratic equation has two real solutions.

Step 1: Identify a, b, and c

From the equation \(2x^2 + 6x - 3 = 0\), the coefficients a, b, and c are as follows: a = 2, b = 6, c = -3.

Step 2: Calculate the discriminant

Now, calculate the discriminant using the formula \(D = b^2 - 4ac\). Substituting the identified values, we get: \(D = 6^2 - 4*2*(-3) = 36 + 24 = 60\).

Step 3: Determine the number of solutions

Since the discriminant D is greater than zero, this means that the quadratic equation has two distinct real solutions.

Key Concepts

Understanding the Discriminant

The discriminant is a special part of the quadratic formula that helps determine the nature and number of solutions for a quadratic equation. A quadratic equation is generally written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients.
The formula for the discriminant \(D\) is \(D = b^2 - 4ac\). By calculating this value, we can predict the type of solutions we might get without actually having to solve the equation.

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, which is sometimes referred to as a repeated or double root.
• If \(D < 0\), no real solutions exist because the roots are complex numbers.

In the given equation \(y=2x^{2}+6x-3\), the discriminant was calculated to be 60, which is positive, indicating two distinct real solutions.

Real Solutions Explained

Real solutions are the values of \(x\) that satisfy the quadratic equation, making the expression equal to zero. When dealing with real solutions, we focus on numbers that are not imaginary or complex.
Real solutions occur when the graph of the quadratic equation, which is a parabola, intersects the x-axis.
Depending on the discriminant, we can have:

• Two distinct real solutions: The parabola crosses the x-axis at two different points. This happens when the discriminant is greater than zero.
• One real solution: The parabola touches the x-axis at just one point (the vertex). This occurs when the discriminant is exactly zero.
• No real solutions: The parabola does not cross the x-axis at all, which means the solutions are not real numbers. This happens when the discriminant is less than zero.

For our equation, because the discriminant is 60, which is greater than zero, we have two distinct real solutions.

Role of Coefficients in Quadratic Equations

Coefficients in a quadratic equation \(ax^2 + bx + c = 0\) play crucial roles in shaping its properties and solutions. Each coefficient represents different elements in the equation.

• \(a\) (Leading coefficient): This determines the parabola's direction. If \(a\) is positive, the parabola opens upward. If negative, downward. It also influences the "width" or "narrowness" of the parabola.
• \(b\) (Linear coefficient): This coefficient affects the position of the parabola, including its vertex along the x-axis. It also influences the symmetry of the parabola.
• \(c\) (Constant term): This is the y-intercept of the parabola, indicating where it crosses the y-axis when \(x=0\).

In the equation \(2x^2 + 6x - 3\), \(a = 2\), \(b = 6\), and \(c = -3\) determine many aspects such as orientation and intersection points, contributing to the calculation of the discriminant, revealing the number and type of solutions.