Problem 61 - Elementary Algebra

Question

\(m^{2}=20 m+6\)

Step-by-Step Solution

Answer

The solutions are \(m_1 = 10 + \sqrt{106}\) and \(m_2 = 10 - \sqrt{106}\).

Step 1 - Write down the equation

Start by writing down the given equation: \[m^{2} = 20m + 6\]

Step 2 - Rearrange the equation

Subtract \(20m\) and \(6\) from both sides to set the equation to zero:\[m^{2} - 20m - 6 = 0\]

Step 3 - Identify coefficients

Identify the coefficients in the quadratic equation \(am^{2} + bm + c = 0\):\[a = 1, \ b = -20, \ c = -6\]

Step 4 - Solve using the quadratic formula

Use the quadratic formula \(m = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substitute the identified coefficients:\[m = \frac{-(-20) \pm \sqrt{(-20)^{2} - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1}\]

Step 5 - Simplify under the square root

First, calculate the discriminant:\[(-20)^{2} - 4 \cdot 1 \cdot (-6) = 400 + 24 = 424\]Now, substitute it back into the formula:\[m = \frac{20 \pm \sqrt{424}}{2}\]

Step 6 - Simplify the complete expression

Simplify the expression inside the square root and the overall fraction:\[\sqrt{424} = 2\sqrt{106}\]Thus the equation becomes:\[m = \frac{20 \pm 2\sqrt{106}}{2}\] Which can then be simplified to:\[m = 10 \pm \sqrt{106}\]

Step 7 - Write the final solutions

Finally, separate the solutions:\[m_1 = 10 + \sqrt{106}\] \[m_2 = 10 - \sqrt{106}\]

Key Concepts

Quadratic Formula

One principal method of solving quadratic equations is the Quadratic Formula. This formula provides the exact solutions for any quadratic equation of the form:

• \(ax^2 + bx + c = 0 \)

To solve this, we use the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's what each term represents:

• \(a \): Coefficient of \(x^2\)
• \(b \): Coefficient of \(x\)
• \(c \): Constant term

This formula is derived from the process of completing the square and provides a straightforward way to find the \(x\) values that satisfy the quadratic equation. Remember to always rearrange your equation in the \(ax^2 + bx + c = 0\) format before applying the formula.

Discriminant

The discriminant helps us understand the nature of the roots (solutions) of a quadratic equation before fully solving it. Found inside the quadratic formula, it is calculated as: \[ b^2 - 4ac \]Here are the critical insights you can gain from the value of the discriminant:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (also called a repeated or double root).
• If \(b^2 - 4ac < 0\), there are no real roots; instead, the roots are complex or imaginary.

In our example, we computed the discriminant as: \[ (-20)^2 - 4 \cdot 1 \cdot (-6) = 400 + 24 = 424 \]Since 424 is greater than 0, it confirms there are two distinct real roots for the equation.

Simplifying Expressions

In many cases, the solutions derived from the quadratic formula may initially appear complicated, requiring further simplification. Let's break down how we simplified it in the given example:

• Start with the expression under the square root: \[ \sqrt{424} = \sqrt{4 \cdot 106} = 2 \sqrt{106} \]
• Then input this simplification back into the quadratic formula: \[ m = \frac{20 \pm 2 \sqrt{106}}{2} \]
• Finally, simplify the overall fraction: \[ m = 10 \pm \sqrt{106} \]

Breaking expressions down into simpler parts makes them more comprehensible and manageable, helping to get the final solutions more clearly: \(m_1 = 10 + \sqrt{106}\) and \(m_2 = 10 - \sqrt{106}\).
Always check if there's a common factor that can be simplified to streamline your solution.