Problem 35 - Elementary Algebra

Question

\(n^{2}-n-56=0\)

Step-by-Step Solution

Answer

The solutions are \( n = 8 \) and \( n = -7 \).

Step 1 - Identify the quadratic equation

Identify the given equation as a quadratic equation in the standard form. The given equation is: \[ n^2 - n - 56 = 0 \]

Step 2 - Recognize coefficients

Identify the coefficients for the quadratic formula where the equation is in the form \[ ax^2 + bx + c = 0 \]. For the given equation, \[ a = 1 \], \[ b = -1 \], and \[ c = -56 \].

Step 3 - Use the quadratic formula

The quadratic formula is given by \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the coefficients: \[ a = 1 \], \[ b = -1 \], and \[ c = -56 \].

Step 4 - Calculate the discriminant

Calculate the discriminant using the formula \[ b^2 - 4ac \]: \[ (-1)^2 - 4(1)(-56) = 1 + 224 = 225 \]

Step 5 - Solve for n

Substitute the discriminant back into the quadratic formula to solve for \( n \): \[ n = \frac{-(-1) \pm \sqrt{225}}{2(1)} = \frac{1 \pm 15}{2} \]

Step 6 - Find the solutions

Calculate the two possible values for \( n \): \[ n = \frac{1 + 15}{2} = 8 \] and \[ n = \frac{1 - 15}{2} = -7 \]

Key Concepts

quadratic formula

The quadratic formula is an important tool when solving quadratic equations. A quadratic equation has the general form ax^2 + bx + c = 0a, b, and c are coefficients. To solve such equations, use the quadratic formula: \[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].The formula provides solutions by calculating the value of n at given coefficient values. For example, if we have the equation \[n^2 - n - 56 = 0\],we can identify a = 1, b = -1, and c = -56. By substituting these values into the quadratic formula, we can find the solutions to the equation.

discriminant

The discriminant is a crucial part of the quadratic formula. It's the part under the square root sign: \[b^2 - 4ac\].The discriminant helps us determine the nature of the roots of a quadratic equation:

• When the discriminant is positive (>b^2 - 4ac > 0), there are two real and distinct solutions.
  
• When the discriminant is zero ( b^2 - 4ac = 0), there is exactly one real solution.
  
• When the discriminant is negative ( b^2 - 4ac < 0), there are no real solutions, only complex ones.
  

For the given equation \[n^2 - n - 56 = 0\], calculate the discriminant: \[ (-1)^2 - 4(1)(-56) = 1 + 224 = 225\].Since 225 is positive, the equation has two real and distinct solutions.

solving quadratic equations

Solving quadratic equations can be done using different methods such as factoring, completing the square, or using the quadratic formula. Using the quadratic formula is a surefire method. Here’s how you solve \[ n^2 - n - 56 = 0\] step-by-step:

• Identify the coefficients: \[a = 1, b = -1, c = -56\].
• Use the quadratic formula: \[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
• Calculate the discriminant: \[b^2 - 4ac\], which is 225.
• Substitute back into the quadratic formula: \[n = \frac{-(-1) \pm \sqrt{225}}{2(1)} = \frac{1 \pm 15}{2}\].
• Solve for nequivalent to:\[\frac{1 + 15}{2} = 8\] and\[\frac{1 - 15}{2} = -7\].

Thus, the equation \[n^2 - n - 56 = 0\]has two solutions n = 8 and n = -7. Understanding these step-by-step methods ensures you can tackle any quadratic equation you encounter!