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Quadratic Equations

1️⃣ Understanding Quadratic Equations (Beyond the Basics)

A quadratic equation is a polynomial equation where the highest power of the variable is 2. It is written in standard form as:

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

  • ✔ \( a, b, c \) are real numbers, and \( a \neq 0 \).
  • ✔ The highest exponent is 2, so the graph is a parabola.

Example:

\[ 3x^2 - 5x + 2 = 0 \]

In Level 1 and Level 2, we covered basic factoring, the quadratic formula, and completing the square. Now, let’s explore advanced methods, applications, and graphical interpretations.

2️⃣ Nature of the Roots – The Role of the Discriminant

The discriminant helps determine the number and type of solutions of a quadratic equation. It is given by:

\[ D = b^2 - 4ac \]

  • ✔ If \( D > 0 \): Two real and distinct roots (parabola crosses the x-axis twice).
  • ✔ If \( D = 0 \): One real repeated root (parabola touches the x-axis at one point).
  • ✔ If \( D < 0 \): No real solutions (parabola does not touch the x-axis).

Example: For \( 4x^2 - 12x + 9 = 0 \):

\[ D = (-12)^2 - 4(4)(9) = 144 - 144 = 0 \]

Since \( D = 0 \), there is one repeated root.

3️⃣ Graphing Quadratic Equations (Beyond the Basics)

Every quadratic equation graphs as a parabola. Key features include:

  • Vertex: The highest or lowest point.
  • Axis of Symmetry: The vertical line through the vertex, given by: \[ x = \frac{-b}{2a} \]
  • X-Intercepts (Roots): Where \( y = 0 \).
  • Y-Intercept: The value of \( y \) when \( x = 0 \).

Example: Graph \( f(x) = x^2 - 6x + 8 \):

Find the vertex:

\( x = \frac{6}{2} = 3 \)

Then, \( f(3) = 9 - 18 + 8 = -1 \); Vertex = \( (3,-1) \).

Find intercepts: Y-intercept \( f(0)=8 \) and by factoring, \( x^2-6x+8=(x-2)(x-4) \) so X-intercepts at \( x=2 \) and \( x=4 \).

The parabola opens upward since \( a=1>0 \).

4️⃣ Advanced Factoring Techniques for Quadratics

Factoring When \( a \neq 1 \) (Decomposition Method)

Example: Factor \( 6x^2 - 7x - 3 \).

Step 1: Multiply \( a \) and \( c \): \( 6 \times (-3) = -18 \).
Step 2: Find two numbers that multiply to -18 and add to -7: (-9 and 2).
Step 3: Rewrite: \( 6x^2 - 9x + 2x - 3 \).
Step 4: Group: \( (6x^2 - 9x) + (2x - 3) \).
Step 5: Factor: \( 3x(2x-3) + 1(2x-3) = (3x+1)(2x-3) \).
Final Answer: \( (3x+1)(2x-3) \).

5️⃣ Solving Quadratics in Real-World Applications

Quadratic equations are used in fields such as physics, business, and engineering. For example, in projectile motion:

Example (Projectile Motion): A ball is thrown upward with an initial velocity of 20 m/s. Its height at time \( t \) seconds is given by:

\[ h(t) = -5t^2 + 20t + 30 \]

Set \( h(t)=0 \) to find when the ball hits the ground, and solve using the quadratic formula:

\( t = \frac{-20 \pm \sqrt{400+600}}{-10} = \frac{-20 \pm \sqrt{1000}}{-10} \)

Final Answer: \( t \approx 5.83 \) seconds.

Practice Questions – Fundamental

  1. Find the number of solutions for: \( 5x^2 - 6x + 2 = 0 \).
  2. Graph \( y = -x^2 + 4x - 3 \) and find its vertex.
  3. Solve using factoring: \( 4x^2 - 9x + 2 = 0 \).
  4. Solve using the quadratic formula: \( x^2 + 5x - 7 = 0 \).
  5. Find the range of: \( f(x) = -2x^2 + 4x + 5 \).
  6. Solve for \( x \) using completing the square: \( x^2 + 10x + 21 = 0 \).
  7. A rectangular garden has an area of 120 m² and its length is 4 m more than its width. Find its dimensions.
  8. The revenue function is \( R(x)=-2x^2+20x \). Find the number of items that maximize revenue.
  9. Find the value of \( k \) for which \( x^2+kx+9 \) has only one solution.
  10. For \( h(t)=-4t^2+16t+50 \), find when the rocket reaches maximum height.

Challenging Problems – Test Your Skills!

  1. Prove that the sum of the roots of \( ax^2+bx+c=0 \) is \( -\frac{b}{a} \) and the product is \( \frac{c}{a} \).
  2. Solve for \( x \): \( \frac{x^2+5x+6}{x+2}=0 \).
  3. Find the equation of a parabola with vertex \( (2,-3) \) passing through \( (0,5) \).
  4. A football is kicked and its height is given by \( h(t)=-5t^2+25t \). Find when the ball reaches maximum height.
  5. Find the integer solutions for \( x^2-7x+12=0 \).

7️⃣ Summary

  • ✅ Quadratic equations are polynomial equations with the highest exponent of 2 and graph as parabolas.
  • ✅ Advanced factoring methods, the quadratic formula, and completing the square are key techniques to solve quadratics.
  • ✅ The discriminant \( D = b^2-4ac \) reveals the number and type of solutions.
  • ✅ Graphing quadratics helps visualize the vertex, axis of symmetry, and intercepts.

8️⃣ Fantastic Job!

Fantastic work! Next up: Graphing Quadratic Functions!

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