Problem 23 - Elementary Algebra

Question

\(m^{2}+8 m-84=0\)

Step-by-Step Solution

Answer

The solutions are \(m = 6\) and \(m = -14\).

Step 1: Identify the quadratic equation

Recognize that the given equation, \(m^2 + 8m - 84 = 0\), is a quadratic equation in the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 8\), and \(c = -84\).

Step 2: Apply the quadratic formula

Use the quadratic formula to find the roots of the equation. The quadratic formula is given by \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the discriminant

Find the discriminant \(\Delta\) using the formula \(\Delta = b^2 - 4ac\). Substitute \(a = 1\), \(b = 8\), and \(c = -84\) into the formula: \(\Delta = 8^2 - 4 \cdot 1 \cdot (-84)\). This simplifies to \(\Delta = 64 + 336 = 400\).

Step 4: Find the square root of the discriminant

Calculate the square root of the discriminant \(\Delta\). \(\sqrt{400} = 20\).

Step 5: Substitute back into the quadratic formula

Substitute \(\Delta\) back into the quadratic formula: \(m = \frac{-8 \pm 20}{2 \cdot 1}\). This yields two solutions: \(m = \frac{-8 + 20}{2}\) and \(m = \frac{-8 - 20}{2}\).

Step 6: Simplify the solutions

Simplify each solution step-by-step: For \(m = \frac{-8 + 20}{2}\), we get \(m = \frac{12}{2} = 6\). For \(m = \frac{-8 - 20}{2}\), we get \(m = \frac{-28}{2} = -14\).

Step 7: Write the final answer

The solutions to the quadratic equation \(m^2 + 8m - 84 = 0\) are \(m = 6\) and \(m = -14\).

Key Concepts

Quadratic Formula

The quadratic formula is used to find the roots of quadratic equations of the form \(ax^2 + bx + c = 0\). It is a crucial tool for solving these equations efficiently. The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
Here is how each part of the formula works:

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