Problem 44 - Elementary Algebra

Question

\(5 k^{2}-k-8=0\)

Step-by-Step Solution

Answer

The solutions are \( k = \frac{1 + \sqrt{161}}{10} \) and \( k = \frac{1 - \sqrt{161}}{10} \).

Step 1 - Identify coefficients

Recognize the standard form of the quadratic equation: It is written as \[ ax^2 + bx + c = 0 \]. Compare with the given equation: \[ 5k^2 - k - 8 = 0 \]. Identify the coefficients: \(a = 5\), \(b = -1\), and \(c = -8\).

Step 2 - Use the quadratic formula

The quadratic formula to find the solutions is: \[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the coefficients \(a\), \(b\), and \(c\) into the formula.

Step 3 - Calculate the discriminant

Compute the discriminant using \[ b^2 - 4ac \]: \( (-1)^2 - 4(5)(-8) = 1 + 160 = 161 \).

Step 4 - Find the square root of the discriminant

Calculate the square root of the discriminant: \( \sqrt{161} \).

Step 5 - Apply the quadratic formula

Plug the values into the quadratic formula: \[ k = \frac{1 \pm \sqrt{161}}{10} \].The solutions are: \[ k = \frac{1 + \sqrt{161}}{10} \] and \[ k = \frac{1 - \sqrt{161}}{10} \].

Key Concepts

coefficients in algebra

Coefficients in algebra are the numerical or constant factors in terms of an equation. When dealing with quadratic equations, which come in the form a x^2 + b x + c = 0, it's crucial to identify these coefficients correctly. In our given exercise, 5 k^2 - k - 8 = 0, the coefficients are:

• a = 5,
• b = -1,
• c = -8.

Recognizing these coefficients is the first step to solve the equation, as they plug into the quadratic formula.

quadratic formula

The quadratic formula is a powerful tool to solve quadratic equations. The formula is written as: \[ k = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]. This formula directly uses the coefficients identified earlier. For our exercise, you need to substitute a = 5, b = -1, c = -8 into this formula. It simplifies the process of solving, no matter how complex the equation looks. First, calculate the discriminant inside the square root.

discriminant calculation

The discriminant \( \Delta \), is found using \( b^2 - 4ac \), and it helps determine the nature of the roots. For our example:\( (-1)^2 - 4 \times 5 \times -8 \)\( 1 + 160 = 161 \). Since \( \Delta \) \( = 161 \) is a positive number, it shows that the quadratic equation has two real and distinct roots. The next step is finding the square root of the discriminant, which is \( \sqrt{161} \).

solving quadratic equations

After calculating the discriminant and substituting back into the quadratic formula, you'll get: \( k = \frac{1 \pm \sqrt{161}}{10} \). This gives you two solutions:

• \( k = \frac{1 + \sqrt{161}}{10} \),
• \( k = \frac{1 - \sqrt{161}}{10} \).

These are the solutions to the quadratic equation 5 k^2 - k - 8 = 0. Understanding each step of using the quadratic formula helps you solve any quadratic equation efficiently. Practice is key to mastering these types of problems.