Problem 35 - Algebra A Combined Function

Question

Use the quadratic formula to solve each equation. These equations have real solutions and complex, but not real, solutions. $$ \frac{2}{5} y^{2}+\frac{1}{5} y+\frac{3}{5}=0 $$

Step-by-Step Solution

Answer

The solutions are complex and given by \( y = \frac{-1 \pm i\sqrt{23}}{4} \).

Step 1: Convert Equation to Standard Form

First, recognize that the equation is already in the quadratic form, but it has fractions which can be cumbersome to work with. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Multiply every term by 5 to eliminate the fractions, resulting in \( 2y^2 + y + 3 = 0 \).

Step 2: Identify Coefficients

Now that the equation is in the standard form \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 2 \), \( b = 1 \), and \( c = 3 \).

Step 3: Set Up the Quadratic Formula

The quadratic formula is \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Use the identified coefficients \( a = 2 \), \( b = 1 \), and \( c = 3 \) in this formula.

Step 4: Calculate the Discriminant

The discriminant is \( b^2 - 4ac \). Substitute the values: \( b = 1 \), \( a = 2 \), and \( c = 3 \). Compute: \( \Delta = 1^2 - 4 \times 2 \times 3 = 1 - 24 = -23 \).

Step 5: Determine Nature of Solutions

Since the discriminant \( \Delta = -23 \) is less than zero, the quadratic equation has two complex but not real solutions.

Step 6: Compute the Complex Solutions

Substitute \( b = 1 \), \( a = 2 \), and \( \Delta = -23 \) into the quadratic formula: \( y = \frac{-1 \pm \sqrt{-23}}{4} \).Simplify the expression using \( \sqrt{-23} = i\sqrt{23} \), leading to:\[ y = \frac{-1 \pm i\sqrt{23}}{4} \].

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. The quadratic formula provides a method to find the values of \( x \) that make the equation true. The formula is expressed as:
\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• "\( -b \pm \sqrt{b^2 - 4ac}\)" finds the values of \( x \)
• "\( 2a \)" represents the division by twice the coefficient of \( x^2 \)

This formula calculates the roots of the quadratic equation, helping us find the solutions. It is important because it allows us to solve any quadratic equation, regardless of whether the solutions are real or complex.
Solving the equation involves substituting the values for \( a \), \( b \), and \( c \) into the formula, simplifying, and identifying the solutions.

Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is less than zero. The solutions are no longer real numbers, and are expressed in terms of the imaginary unit, \( i \), where \( i^2 = -1 \).
For example, when we have a discriminant of -23, the solutions involve the square root of a negative number which is expressed as:
\[y = \frac{-b \pm \sqrt{-23}}{2a}\]This simplifies to:

• \( y = \frac{-1 \pm i\sqrt{23}}{4} \)

In this form, "\(i\sqrt{23}\)" represents the imaginary part of the solution. While real solutions only involve numbers without "i," complex solutions have both real and imaginary components. Complex solutions are very important in mathematics because they extend our understanding of numbers and solutions but aren't visible on a standard number line like real numbers.

Discriminant

The discriminant of a quadratic equation is a key feature that helps determine the nature of its solutions. The discriminant is calculated as:
\[\Delta = b^2 - 4ac\]

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), the equation has exactly one real solution (or a repeated root).
• If \( \Delta < 0 \), the equation has two complex solutions.

In the case of the equation \( ax^2 + bx + c = 0 \) with coefficients \( a = 2 \), \( b = 1 \), and \( c = 3 \), the discriminant is:
\( \Delta = 1^2 - 4 \times 2 \times 3 = -23 \)
Since \( \Delta \) is negative, the solutions are complex. Calculating the discriminant before solving an equation gives us insight into what type of solutions to expect.

Coefficients

Coefficients are the numerical factors in the terms of a polynomial or equation. For the quadratic equation \( ax^2 + bx + c = 0 \),

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

Understanding the role of coefficients is crucial because they directly affect the shape and position of the parabola represented by the quadratic equation, and they are used in calculating the solutions through the quadratic formula.
In solving the given equation, \( 2y^2 + y + 3 = 0 \), identifying \( a = 2 \), \( b = 1 \), and \( c = 3 \) allows us to substitute these values efficiently into the quadratic formula and discriminant calculation. The coefficients help define the specific properties of a quadratic equation and are essential for understanding and solving the equation.