Problem 36 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ d=4+\frac{21}{d} $$

Step-by-Step Solution

Answer

The solutions are \( d = 7 \) and \( d = -3 \).

Step 1: Understand the Equation

We begin with the equation \( d = 4 + \frac{21}{d} \). The task is to solve for \( d \). This is a rational equation because it involves a variable in the denominator.

Step 2: Eliminate the Fraction

To solve the equation, we need to eliminate the fraction. Multiply both sides of the equation by \( d \) to get:\[ d^2 = 4d + 21 \]

Step 3: Rearrange into a Quadratic Equation

Rearrange the terms to form a standard quadratic equation:\[ d^2 - 4d - 21 = 0 \]

Step 4: Solve the Quadratic Equation

To solve \( d^2 - 4d - 21 = 0 \), we use the quadratic formula:\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this equation, \( a = 1 \), \( b = -4 \), and \( c = -21 \).

Step 5: Calculate the Discriminant and Roots

Calculate the discriminant:\[ b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-21) = 16 + 84 = 100 \]Since the discriminant is positive, there are two real roots.Calculate the roots using the quadratic formula:\[ d = \frac{-(-4) \pm \sqrt{100}}{2 \times 1} = \frac{4 \pm 10}{2} \]The roots are \( d = \frac{14}{2} = 7 \) and \( d = \frac{-6}{2} = -3 \).

Step 6: Check the Solutions

Substitute \( d = 7 \) back into the original equation:\( 7 = 4 + \frac{21}{7} \) -> \( 7 = 4 + 3 \) -> \( 7 = 7 \). This is correct.Substitute \( d = -3 \) back into the original equation:\( -3 = 4 + \frac{21}{-3} \) -> \( -3 = 4 - 7 \) -> \( -3 = -3 \). This is also correct.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions where the numerator, the denominator, or both contain a variable. They can often seem intimidating because of these fractional parts. To solve rational equations, we typically clear the fractions by multiplying through by a common denominator. This step aims to transform the equation into one without fractions, thus simplifying it. For example, in the original equation, \( d = 4 + \frac{21}{d} \), the denominator \( d \) gives it its rational equation nature.

• First, we identify that a fraction exists due to a variable in the denominator.
• Then, we eliminate the fraction by multiplying both sides by this variable, which simplifies our work.

This method helps in transitioning the problem into a more recognizable form, sometimes involving a quadratic equation, which is easier to solve.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to solve for the values of \( x \), regardless of how messy the equation appears. The quadratic formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our scenario, after converting the rational equation to a quadratic form \( d^2 - 4d - 21 = 0 \), the coefficients \( a = 1 \), \( b = -4 \), and \( c = -21 \) are used in this formula.

• By substituting these values into the formula, it helps find the roots at once.
• This method bypasses trial and error, making it especially useful when factoring is difficult or impossible.

Applying the quadratic formula can be easy once you identify and plug in the correct values.

Discriminant

The discriminant is a component of the quadratic formula found under the square root: \( b^2 - 4ac \). It plays a critical role in determining the nature and number of the roots of the quadratic equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, meaning the parabola touches the x-axis at one point.
• If the discriminant is negative, the equation has no real roots and the solutions are complex numbers.

For this particular problem, the discriminant calculation is:\[ (-4)^2 - 4 \times 1 \times (-21) = 16 + 84 = 100 \]This positive result tells us that the equation has two real roots, neatly demonstrating the importance of the discriminant in predicting solution outcomes.

Real Roots

Real roots are the solutions of a quadratic equation that are real numbers as opposed to imaginary numbers. They represent the x-intercepts of the parabola described by the quadratic equation when graphed on a coordinate plane. In the context of solving quadratic equations, determining the nature of these roots is vital.

• The roots give us important information about where the function intersects or "touches" the x-axis.
• This is crucial for many applications in science and engineering, where solutions must be real, measurable quantities.

From the quadratic equation \( d^2 - 4d - 21 = 0 \), the calculated roots were \( d = 7 \) and \( d = -3 \). These values satisfy the original equation and can be verified by substitution back into the initial equation. Both values resolve correctly, affirming their validity as real roots.