Quadratic Equations
1️⃣ What is a Quadratic Equation?
A quadratic equation is an equation where the highest exponent of the variable is 2. Unlike linear equations, which form straight lines, quadratic equations always graph as a parabola (a U-shaped curve).
🔹 Standard Form of a Quadratic Equation
A quadratic equation is written as:
\[ ax^2 + bx + c = 0 \]
- ✔ \( a \), \( b \), and \( c \) are constants (numbers).
- ✔ \( x \) is the variable (the unknown we solve for).
- ✔ \( a \neq 0 \) (if \( a = 0 \), the equation becomes linear).
Examples of Quadratic Equations:
- \( x^2 - 5x + 6 = 0 \)
- \( 2x^2 + 3x - 7 = 0 \)
- \( -4x^2 + 9x + 1 = 0 \)
Not Quadratic Equations:
- \( x^3 - 2x + 5 = 0 \) (cubic equation)
- \( \frac{1}{x^2} + 3x = 4 \) (variable in denominator)
- \( x + 2 = 0 \) (linear equation)
2️⃣ Understanding the Graph of a Quadratic Equation
Every quadratic equation graphs as a parabola.
- ✔ If \( a > 0 \) → the parabola opens upwards (like a smile 😀).
- ✔ If \( a < 0 \) → the parabola opens downwards (like a frown 🙁).
🌟 Key Features of a Parabola
- Vertex – The highest or lowest point of the parabola.
- Axis of Symmetry – A vertical line dividing the parabola into two equal halves.
- X-Intercepts (Roots) – Points where the parabola crosses the x-axis (\( y = 0 \)).
- Y-Intercept – The point where the parabola crosses the y-axis (\( x = 0 \)).
Example: For \( y = x^2 - 4x + 3 \), the parabola crosses the x-axis at \( x = 1 \) and \( x = 3 \) (these are the roots).
3️⃣ How to Solve Quadratic Equations
1️⃣ Factoring Method (Easiest Method if it Works!)
Example: Solve \( x^2 - 5x + 6 = 0 \).
Step 1: Find two numbers that multiply to 6 and add to -5 (they are -2 and -3).
Step 2: Write the factors: \( (x - 2)(x - 3) = 0 \).
Step 3: Solve: \( x - 2 = 0 \) or \( x - 3 = 0 \).
Final Answer: \( x = 2, \; x = 3 \).
2️⃣ The Quadratic Formula (Works for Any Quadratic!)
Example: Solve \( 2x^2 - 3x - 5 = 0 \).
Step 1: Identify \( a=2 \), \( b=-3 \), \( c=-5 \).
Step 2: Use the formula:
\[
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
\]
\( x = \frac{3 \pm \sqrt{9+40}}{4} = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4} \).
Step 3: Solve: \( x = \frac{10}{4} = 2.5 \) or \( x = \frac{-4}{4} = -1 \).
Final Answer: \( x = 2.5, \; x = -1 \).
3️⃣ Completing the Square (Useful for Certain Problems!)
Example: Solve \( x^2 - 6x + 5 = 0 \).
Step 1: Rewrite as \( x^2 - 6x = -5 \).
Step 2: Add \( \left(\frac{-6}{2}\right)^2 = 9 \) to both sides: \( x^2 - 6x + 9 = 4 \).
Step 3: Write as \( (x - 3)^2 = 4 \).
Step 4: Take the square root: \( x - 3 = \pm 2 \).
Final Answer: \( x = 3 \pm 2 \) → \( x = 5 \) or \( x = 1 \).
4️⃣ The Discriminant (How Many Solutions?)
The discriminant is given by:
\[ D = b^2 - 4ac \]
- If \( D > 0 \): Two real solutions.
- If \( D = 0 \): One real solution.
- If \( D < 0 \): No real solutions.
Example: For \( x^2 - 4x + 4 = 0 \), \( D = (-4)^2 - 4(1)(4) = 16 - 16 = 0 \).
Only one real solution!
5️⃣ Fancy Practice Questions 🎯
Basic Practice – Let’s Get Started!
- Solve \( x^2 - 5x + 6 = 0 \) by factoring.
- Solve \( x^2 + 4x - 7 = 0 \) using the quadratic formula.
- Solve \( x^2 - 4x + 3 = 0 \) by completing the square.
- Find the vertex of \( y = x^2 - 6x + 7 \).
- Determine the number of solutions for \( 3x^2 - 5x + 2 = 0 \).
- Graph \( y = -2x^2 + 4x - 1 \).
Challenging Questions – Test Your Skills!
- A projectile follows \( h = -5t^2 + 20t + 30 \). When does it reach its highest point?
- Solve \( 2x^2 - 3x + 1 = 0 \) using both the quadratic formula and completing the square. Compare the results.
- A bridge has a parabolic arch \( y = -x^2 + 8x - 15 \). Find where it touches the ground.
- A ball is thrown upward with \( h = -4.9t^2 + 20t + 1.5 \). Find when it hits the ground.
- Find the equation of a parabola with vertex \( (3,2) \) passing through \( (5,10) \).
- The sum of a number and its square is 12. Find the number.
6️⃣ Summary
- ✅ Quadratic equations graph as parabolas.
- ✅ They can be solved by factoring, using the quadratic formula, or completing the square.
- ✅ The discriminant \( D = b^2 - 4ac \) indicates the number of real solutions.
- ✅ Graphing quadratics helps visualize the vertex, axis of symmetry, and intercepts.
7️⃣ Great Job!
Awesome work! You're mastering quadratics. Next up: Systems of Linear Equations!