Worked solutions for completing the square in parabola equations

📚 Understanding Completing the Square for Parabolas

Completing the square is a technique used to rewrite a quadratic expression in the form $ax^2 + bx + c$ into the form $a(x - h)^2 + k$, which directly reveals the vertex $(h, k)$ of the parabola. This is incredibly useful for graphing and analyzing quadratic functions.

📜 Historical Context

The concept of completing the square dates back to ancient Babylonian mathematicians who used geometric methods to solve quadratic equations. The algebraic formulation we use today evolved over centuries, becoming a standard technique in algebra by the Renaissance.

🔑 Key Principles

• ➕ Leading Coefficient: Ensure the coefficient of $x^2$ is 1. If not, factor it out from the $x^2$ and $x$ terms.
• ➗ Halving the Coefficient: Take half of the coefficient of the $x$ term (the 'b' value).
• 🧮 Squaring: Square the result from the previous step.
• ➕ Adding and Subtracting: Add and subtract this squared value within the expression to maintain the equation's balance.
• ✍️ Factoring: Factor the perfect square trinomial.
• 💡 Simplifying: Simplify the expression to the vertex form.

✏️ Example 1: Basic Completing the Square

Rewrite $x^2 + 6x + 5$ in vertex form.

1. The coefficient of $x^2$ is already 1.
2. Half of 6 is 3.
3. 3 squared is 9.
4. Add and subtract 9: $x^2 + 6x + 9 - 9 + 5$
5. Factor: $(x + 3)^2 - 9 + 5$
6. Simplify: $(x + 3)^2 - 4$

The vertex is $(-3, -4)$.

🧪 Example 2: Completing the Square with $a \neq 1$

Rewrite $2x^2 - 8x + 10$ in vertex form.

1. Factor out 2: $2(x^2 - 4x) + 10$
2. Half of -4 is -2.
3. (-2) squared is 4.
4. Add and subtract 4 inside the parentheses: $2(x^2 - 4x + 4 - 4) + 10$
5. Factor: $2[(x - 2)^2 - 4] + 10$
6. Distribute and simplify: $2(x - 2)^2 - 8 + 10 = 2(x - 2)^2 + 2$

The vertex is $(2, 2)$.

📈 Example 3: Dealing with Fractions

Rewrite $x^2 + 3x - 1$ in vertex form.

1. The coefficient of $x^2$ is already 1.
2. Half of 3 is $\frac{3}{2}$.
3. $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$.
4. Add and subtract $\frac{9}{4}$: $x^2 + 3x + \frac{9}{4} - \frac{9}{4} - 1$
5. Factor: $\left(x + \frac{3}{2}\right)^2 - \frac{9}{4} - 1$
6. Simplify: $\left(x + \frac{3}{2}\right)^2 - \frac{13}{4}$

The vertex is $\left(-\frac{3}{2}, -\frac{13}{4}\right)$.

🤔 Example 4: Advanced Scenario

Rewrite $-3x^2 - 12x - 5$ in vertex form.

1. Factor out -3: $-3(x^2 + 4x) - 5$
2. Half of 4 is 2.
3. 2 squared is 4.
4. Add and subtract 4 inside the parentheses: $-3(x^2 + 4x + 4 - 4) - 5$
5. Factor: $-3[(x + 2)^2 - 4] - 5$
6. Distribute and simplify: $-3(x + 2)^2 + 12 - 5 = -3(x + 2)^2 + 7$

The vertex is $(-2, 7)$.

📝 Practice Quiz

Rewrite the following quadratic equations in vertex form:

1. ❓ $x^2 + 2x + 3$
2. ❓ $2x^2 + 8x - 5$
3. ❓ $x^2 - 5x + 2$
4. ❓ $-x^2 + 6x + 1$
5. ❓ $3x^2 - 12x + 7$
6. ❓ $x^2 + x - 1$
7. ❓ $-2x^2 - 4x + 3$

✅ Solutions

1. ✔️ $(x + 1)^2 + 2$
2. ✔️ $2(x + 2)^2 - 13$
3. ✔️ $\left(x - \frac{5}{2}\right)^2 - \frac{17}{4}$
4. ✔️ $-(x - 3)^2 + 10$
5. ✔️ $3(x - 2)^2 - 5$
6. ✔️ $\left(x + \frac{1}{2}\right)^2 - \frac{5}{4}$
7. ✔️ $-2(x + 1)^2 + 5$

🌍 Real-World Applications

• 🌉 Engineering: Designing parabolic arches in bridges.
• 📡 Physics: Modeling projectile motion.
• 🛰️ Astronomy: Analyzing parabolic orbits of comets.
• 📈 Economics: Optimizing profit functions.

💡 Conclusion

Completing the square is a powerful technique with diverse applications. Mastering it provides a deeper understanding of quadratic functions and their properties. Keep practicing, and you'll become proficient in no time!