Problem 38 - College Algebra

Question

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 2 x^{2}+5 x+3=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = -1 \) and \( x = -\frac{3}{2} \).

Step 1: Identify A, B, and C

In the quadratic equation form \( ax^2 + bx + c = 0 \), identify the values of \( a \), \( b \), and \( c \). For the equation \( 2x^2 + 5x + 3 = 0 \), we have \( a = 2 \), \( b = 5 \), and \( c = 3 \).

Step 2: Use the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 2 \), \( b = 5 \), and \( c = 3 \) into the formula.

Step 3: Calculate the Discriminant

Calculate the discriminant, \( b^2 - 4ac \). For our equation, it is \( (5)^2 - 4(2)(3) = 25 - 24 = 1 \).

Step 4: Determine the Nature of the Roots

Since the discriminant (\( b^2 - 4ac = 1 \)) is positive, the quadratic equation has two distinct real solutions.

Step 5: Solve for the Roots

Substitute the values into the quadratic formula: \( x = \frac{-5 \pm \sqrt{1}}{4} \). This simplifies to \( x = \frac{-5 + 1}{4} \) and \( x = \frac{-5 - 1}{4} \).

Step 6: Simplify the Solutions

Calculate each solution: \( x = \frac{-5 + 1}{4} = \frac{-4}{4} = -1 \) and \( x = \frac{-5 - 1}{4} = \frac{-6}{4} = -\frac{3}{2} \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. In general form, it is written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The equation is called "quadratic" because the highest power of the variable \( x \) is two.
Quadratic equations are fundamental in algebra because they represent parabolas when graphed on the coordinate plane. Solving these equations is crucial, as they appear in various areas of science and engineering. Methods to solve quadratic equations include factoring, completing the square, and the quadratic formula.
In the provided exercise, the quadratic equation is \( 2x^2 + 5x + 3 = 0 \). Identifying the coefficients here gives us \( a = 2 \), \( b = 5 \), and \( c = 3 \). This foundational understanding is the first step to finding the solutions of the quadratic equation.

Discriminant

The discriminant is an important concept in solving quadratic equations, especially when using the quadratic formula. It is part of the expression under the square root: \( b^2 - 4ac \). This value helps determine the nature of the roots of a quadratic equation.
In the example provided, the discriminant is calculated as \( (5)^2 - 4(2)(3) = 25 - 24 = 1 \). The discriminant can be:

• Positive: The equation has two distinct real solutions.
• Zero: The equation has exactly one real solution, also known as a double root.
• Negative: The equation has no real solutions; the solutions are complex numbers.

Understanding the discriminant is crucial for anticipating the nature of the solutions before solving the equation entirely.

Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation, resulting in real numbers. These solutions are derived from the quadratic formula and related directly to the discriminant.
When the discriminant is positive, as in our example where it equals 1, it indicates that the equation has two distinct real solutions. These real solutions can be found by solving the equation \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
It's important to understand that real solutions are graphically represented by the points where the parabola intersects the x-axis of a graph. For the equation \( 2x^2 + 5x + 3 = 0 \), the solutions are the \( x \)-intercepts of the parabola plotted for this quadratic.

Roots

Roots of a quadratic equation are the solutions obtained by setting the quadratic expression equal to zero. These solutions can be found using various methods depending on the coefficients and the discriminant.
The roots signify the values of \( x \) at which the quadratic equation equals zero, representing the points where the graph of the quadratic function intersects the x-axis.
In this exercise, the roots of the equation \( 2x^2 + 5x + 3 = 0 \) were found using the quadratic formula:

• \( x = \frac{-5 + \sqrt{1}}{4} = -1 \)
• \( x = \frac{-5 - \sqrt{1}}{4} = -\frac{3}{2} \)

These roots mean that when \( x = -1 \) and \( x = -\frac{3}{2} \), the quadratic expression equals zero. Recognizing roots is vital for solving equations and interpreting the geometric representation of quadratic functions.