Problem 40 - Elementary Algebra

Question

(a) solve. (b) check. $$ n^{2}-n-30=0 $$

Step-by-Step Solution

Answer

The solutions are n = 6 and n = -5.

Step 1 - Write Down the Quadratic Equation

Given the quadratic equation: \[ n^{2} - n - 30 = 0 \]

Step 2 - Identify Coefficients

Identify the coefficients from the quadratic equation of the form \(an^2 + bn + c = 0\): \(a = 1\), \(b = -1\), \(c = -30\)

Step 3 - Apply the Quadratic Formula

The quadratic formula is given by: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values of \(a\), \(b\), and \(c\): \[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-30)}}{2(1)} \] This simplifies to: \[ n = \frac{1 \pm \sqrt{1 + 120}}{2} \] \[ n = \frac{1 \pm \sqrt{121}}{2} \] \[ n = \frac{1 \pm 11}{2} \]

Step 4 - Solve for the Roots

Now calculate the roots separately: \[ n = \frac{1 + 11}{2} = \frac{12}{2} = 6 \] \[ n = \frac{1 - 11}{2} = \frac{-10}{2} = -5 \]

Step 5 - Verify the Solutions

Substitute the solutions back into the original equation to verify: For \( n = 6 \): \[ 6^2 - 6 - 30 = 36 - 6 - 30 = 0 \] For \( n = -5 \): \[ (-5)^2 - (-5) - 30 = 25 + 5 - 30 = 0 \]

Key Concepts

Quadratic formula

The quadratic formula is essential for solving quadratic equations of the form: \(ax^2 + bx + c = 0\). The formula helps to find the roots (solutions) of the equation and is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
In this equation:

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

The part inside the square root, \(b^2 - 4ac\), is called the discriminant. The discriminant helps to determine the nature of the roots. If it is positive, there are two distinct real roots. If zero, there is exactly one real root. If negative, there are no real roots.

Identifying coefficients

Before applying the quadratic formula, it's important to identify the coefficient values \(a\), \(b\), and \(c\) in the quadratic equation. Let's look at the equation: \(n^2 - n - 30 = 0\). We can compare this with the standard form \(an^2 + bn + c = 0\) to get our coefficients:

• \(a = 1\), since \(n^2\) is the term with a coefficient of 1
• \(b = -1\), since \(-n\) implies the coefficient is -1
• \(c = -30\), which is the constant term

Identifying these coefficients correctly is a crucial step for applying the quadratic formula accurately.

Solving quadratic equations

With the quadratic formula and identified coefficients, let's solve the quadratic equation:
Using our equation \(n^2 - n - 30 = 0\), the coefficients are: \(a = 1\), \(b = -1\), \(c = -30\). Substituting these into the quadratic formula: \[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-30)}}{2(1)} \] This simplifies to: \[ n = \frac{1 \pm \sqrt{1 + 120}}{2} \] \[ n = \frac{1 \pm \sqrt{121}}{2} \] \[ n = \frac{1 \pm 11}{2} \]
Now, solving for the two values of \(n\):

• \(n = \frac{1 + 11}{2} = 6\)
• \(n = \frac{1 - 11}{2} = -5\)

Verifying solutions

After solving the quadratic equation, it is crucial to verify both solutions by substituting them back into the original equation to ensure they satisfy it. Let's check both solutions:
For \(n = 6\): \[ 6^2 - 6 - 30 = 36 - 6 - 30 = 0 \] This confirms that \(n = 6\) is a valid solution.
For \(n = -5\): \[ (-5)^2 - (-5) - 30 = 25 + 5 - 30 = 0 \] This confirms that \(n = -5\) is also a valid solution.
Thus, both values \(n = 6\) and \(n = -5\) satisfy the equation \(n^2 - n - 30 = 0\). Verifying ensures accuracy and validates the solutions obtained through the quadratic formula.