Problem 61 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ 3 x=5 x^{2}+1 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3 + \sqrt{11}i}{10}\) and \(x = \frac{3 - \sqrt{11}i}{10}\).

Step 1: Arrange the Equation

The given equation is \(3x = 5x^2 + 1\). First, we want to set the equation to 0. To do this, subtract \(3x\) from both sides to get \(0 = 5x^2 + 1 - 3x\). Rearrange the equation to its standard quadratic form: \(5x^2 - 3x + 1 = 0\).

Step 2: Identify Coefficients

Now, identify the coefficients for the quadratic equation \(5x^2 - 3x + 1 = 0\). Here, \(a = 5\), \(b = -3\), and \(c = 1\).

Step 3: Apply the Quadratic Formula

To solve the quadratic equation, use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 4: Calculate the Discriminant

The discriminant \(D\) is given by \(b^2 - 4ac\). Substituting the values gives: \((-3)^2 - 4(5)(1) = 9 - 20 = -11\). Since the discriminant is negative, the solutions are complex numbers.

Step 5: Solve for x

Since the discriminant is negative, the formula \(x = \frac{-b \pm \sqrt{-11}i}{2a}\) is used. Substitute the coefficients: \(x = \frac{-(-3) \pm \sqrt{-11}i}{2(5)} = \frac{3 \pm \sqrt{11}i}{10}\). Thus, the solutions are \(x = \frac{3 + \sqrt{11}i}{10}\) and \(x = \frac{3 - \sqrt{11}i}{10}\).

Key Concepts

Standard Form

The standard form of a quadratic equation is a fundamental concept of quadratic equations. It represents a quadratic equation in the format of \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. This arrangement is crucial because it allows us to apply different methods for finding the solutions or roots of the equation.
In the given exercise, we start with an equation \( 3x = 5x^2 + 1 \). To convert it into standard form, we first rearrange all terms to one side of the equation, resulting in \( 5x^2 - 3x + 1 = 0 \).
This transformation is critical as it sets the stage for applying the quadratic formula and other solving methods effectively.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It provides us a way to find solutions of a quadratic equation directly. The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula applies to any quadratic equation in standard form \( ax^2 + bx + c = 0 \). By substituting the coefficients \( a \), \( b \), and \( c \) from the equation, we can calculate the roots.
In our exercise, the coefficients are \( a = 5 \), \( b = -3 \), and \( c = 1 \). By plugging these values into the quadratic formula, one can solve for \( x \).

Complex Numbers

Complex numbers come into play when the discriminant of a quadratic equation is negative. A complex number has a real part and an imaginary part, denoted by \( a + bi \), where \( i \) is the imaginary unit \( \sqrt{-1} \).
In this exercise, the discriminant calculated was \(-11\), which is negative. Therefore, the solutions of the equation involve complex numbers. Using the quadratic formula, the solutions are presented in the form \( x = \frac{3 \pm \sqrt{11}i}{10} \).
These solutions are known as complex conjugates, which is a common result when dealing with quadratics that do not cross the x-axis.

Discriminant

The discriminant of a quadratic equation is a part of the quadratic formula under the square root, expressed as \( b^2 - 4ac \). It helps determine the nature of the roots:

• If positive, two distinct real roots exist.
• If zero, one real root (or a repeated real root) exists.
• If negative, two complex roots exist.

In the provided example, the discriminant was calculated to be \(-11\). This negative value alerted us to the presence of complex numbers in the solutions.
Understanding the value and significance of the discriminant helps predict the type of solutions we can expect without even calculating them.