Problem 23 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ 25 k^{2}+1=10 k $$

Step-by-Step Solution

Answer

The solution is \( k = \frac{1}{5} \).

Step 1: Rewrite in Standard Form

The first step to solving a quadratic equation is to ensure it is in standard form, which is \( ax^2 + bx + c = 0 \). For the given equation \( 25k^2 + 1 = 10k \), we need to move all terms to one side. We do this by subtracting \(10k\) from both sides:\[ 25k^2 - 10k + 1 = 0 \]

Step 2: Use the Quadratic Formula

For quadratic equations in the form \( ax^2 + bx + c = 0 \), the solutions for \( k \) can be found using the quadratic formula:\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( a = 25 \), \( b = -10 \), and \( c = 1 \). Substitute these values into the formula.

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the solutions:\[ b^2 - 4ac = (-10)^2 - 4 \times 25 \times 1 = 100 - 100 = 0 \]Since the discriminant is 0, there will be exactly one real solution.

Step 4: Solve for k

Substituting into the quadratic formula, we simplify:\[ k = \frac{-(-10) \pm \sqrt{0}}{2 \times 25} = \frac{10 \pm 0}{50} \]This simplifies to:\[ k = \frac{10}{50} = \frac{1}{5} \]

Step 5: Check the Answer

To verify the solution, substitute \( k = \frac{1}{5} \) back into the original equation:\[ 25\left(\frac{1}{5}\right)^2 + 1 = 10\left(\frac{1}{5}\right) \]\[ 25\times\frac{1}{25} + 1 = 2 \]\[ 1 + 1 = 2 \]The original equation is satisfied, confirming that \( k = \frac{1}{5} \) is correct.

Key Concepts

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. A quadratic equation is an equation in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. To find the solutions to this equation, we can use the quadratic formula:

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

This formula always works, regardless of whether the quadratic can be factored or not. It directly utilizes the coefficients of the equation:

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

To solve a quadratic using this formula, identify \( a \), \( b \), and \( c \) from your equation. Substitute these values into the formula and simplify. This yields the solutions for \( x \). They are the roots or zeroes of the equation, the values where the graph touches the x-axis.

Discriminant

The discriminant is a part of the quadratic formula that helps us understand the nature of the roots without actually solving the equation. It is given by \( b^2 - 4ac \). The value of the discriminant tells us:

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution (also called a repeated or double root).
• If \( b^2 - 4ac < 0 \), there are no real solutions, only complex ones.

In the context of the exercise, since the discriminant was calculated to be 0, we found that there is exactly one real solution. This simplifies solving as we don’t have to consider multiple roots, making our job a bit easier. Consider the discriminant as a quick "peek" at what to expect before diving into calculation with the quadratic formula.

Checking Solutions

Checking solutions in a quadratic equation is an essential step to make sure that your calculation is correct. It involves substituting your found solution back into the original equation to verify that both sides indeed equal. Here's how you do it:

• Insert the solution value into every instance of the variable in the original equation.
• Simplify both sides of the equation.
• See if both sides are equal.

If both sides are equal, the solution is correct. In the given exercise, by substituting \( k = \frac{1}{5} \) back into the equation \( 25k^2 + 1 = 10k \), we verified that it satisfies the equation, confirming it as the correct root.By thoroughly checking, you verify not only computation accuracy but also reinforces understanding and confidence in using mathematical procedures to solve quadratic equations.