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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).

πŸ”Ή Standard Form of a Quadratic Equation

A quadratic equation is written as:

\[ ax^2 + bx + c = 0 \]

  • βœ” \( a \), \( b \), and \( c \) are constants (numbers).
  • βœ” \( x \) is the variable (the unknown we solve for).
  • βœ” \( a \neq 0 \) (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.

  • βœ” If \( a > 0 \) β†’ the parabola opens upwards (like a smile πŸ˜€).
  • βœ” If \( a < 0 \) β†’ the parabola opens downwards (like a frown πŸ™).

🌟 Key Features of a Parabola

  1. Vertex – The highest or lowest point of the parabola.
  2. Axis of Symmetry – A vertical line dividing the parabola into two equal halves.
  3. X-Intercepts (Roots) – Points where the parabola crosses the x-axis (\( y = 0 \)).
  4. 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: \( x = 3 \pm 2 \) β†’ \( x = 5 \) 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!

  1. Solve \( x^2 - 5x + 6 = 0 \) by factoring.
  2. Solve \( x^2 + 4x - 7 = 0 \) using the quadratic formula.
  3. Solve \( x^2 - 4x + 3 = 0 \) by completing the square.
  4. Find the vertex of \( y = x^2 - 6x + 7 \).
  5. Determine the number of solutions for \( 3x^2 - 5x + 2 = 0 \).
  6. Graph \( y = -2x^2 + 4x - 1 \).

Challenging Questions – Test Your Skills!

  1. A projectile follows \( h = -5t^2 + 20t + 30 \). When does it reach its highest point?
  2. Solve \( 2x^2 - 3x + 1 = 0 \) using both the quadratic formula and completing the square. Compare the results.
  3. A bridge has a parabolic arch \( y = -x^2 + 8x - 15 \). Find where it touches the ground.
  4. A ball is thrown upward with \( h = -4.9t^2 + 20t + 1.5 \). Find when it hits the ground.
  5. Find the equation of a parabola with vertex \( (3,2) \) passing through \( (5,10) \).
  6. The sum of a number and its square is 12. Find the number.

6️⃣ Summary

  • βœ… Quadratic equations graph as parabolas.
  • βœ… They can be solved by factoring, using the quadratic formula, or completing the square.
  • βœ… The discriminant \( D = b^2 - 4ac \) indicates the number of real solutions.
  • βœ… 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!

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