Describe how to derive the quadratic formula from a quadratic equation in standard form.

Deriving the quadratic formula involves manipulating the standard form of a quadratic equation to solve for the variable \( x \). Here's a step-by-step guide to deriving the quadratic formula from the standard quadratic equation: ### **Standard Form of a Quadratic Equation** A quadratic equation in standard form is: \[ ax^2 + bx + c = 0 \] where: - \( a \), \( b \), and \( c \) are coefficients with \( a \neq 0 \) - \( x \) represents the variable ### **Goal** Solve for \( x \) in terms of \( a \), \( b \), and \( c \). ### **Step 1: Divide All Terms by \( a \)** To simplify, divide every term in the equation by \( a \) (since \( a \neq 0 \)): \[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \] This simplifies the equation to: \[ x^2 + \frac{b}{a}x = -\frac{c}{a} \] ### **Step 2: Complete the Square** The idea is to convert the left side of the equation into a perfect square trinomial. 1. **Identify the Coefficient of \( x \):** The coefficient of \( x \) is \( \frac{b}{a} \). 2. **Find Half of This Coefficient:** \[ \frac{1}{2} \cdot \frac{b}{a} = \frac{b}{2a} \] 3. **Square This Half-Coefficient:** \[ \left( \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2} \] 4. **Add and Subtract This Square Inside the Equation:** \[ x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = \frac{b^2}{4a^2} - \frac{c}{a} \] This step ensures that the equation remains balanced. ### **Step 3: Rewrite the Left Side as a Perfect Square** The left side of the equation now forms a perfect square trinomial: \[ \left( x + \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2} - \frac{c}{a} \] ### **Step 4: Simplify the Right Side** Combine the terms on the right side by finding a common denominator: \[ \frac{b^2}{4a^2} - \frac{4ac}{4a^2} = \frac{b^2 - 4ac}{4a^2} \] So the equation becomes: \[ \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2} \] ### **Step 5: Take the Square Root of Both Sides** Apply the square root to both sides, remembering to consider both the positive and negative roots: \[ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \] ### **Step 6: Solve for \( x \)** Isolate \( x \) by subtracting \( \frac{b}{2a} \) from both sides: \[ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \] ### **Step 7: Combine the Terms Over a Common Denominator** Since both terms have the same denominator, they can be combined: \[ x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a } \] ### **Final Quadratic Formula** \[ x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a } \] ### **Summary** By completing the square on the standard form of the quadratic equation \( ax^2 + bx + c = 0 \), we've derived the quadratic formula, which provides the solutions for \( x \) in terms of the coefficients \( a \), \( b \), and \( c \). This formula is invaluable for solving any quadratic equation, especially when factoring is difficult or impossible.