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Which subject	Mathematics
What age group	Year or Grade 11
What topic	quadratic equations
Number of slides	7
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Introduction to Quadratic Equations

• Definition: A quadratic equation is a polynomial equation of the form ( ax^2 + bx + c = 0 ), where ( a \neq 0 ).
• Characteristics: The highest degree of the variable ( x ) is 2, and the graph of a quadratic equation is a parabola.
• Applications: Used in various fields such as physics (projectile motion), engineering (designs), and finance (profit analysis).

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Standard Form of a Quadratic Equation

• The standard form: ( ax^2 + bx + c = 0 )
• Coefficients:
  • ( a ) (leading coefficient): determines the direction of the parabola (opens upward if ( a > 0 ) and downward if ( a < 0 )).
  • ( b ): affects the position of the vertex on the x-axis.
  • ( c ): the y-intercept of the parabola.
• Example: Consider ( 2x^2 - 4x + 1 = 0 ).

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Factoring Quadratic Equations

• Definition: Expressing ( ax^2 + bx + c ) as ( (px + q)(rx + s) = 0 ).
• Steps:
  1. Set the equation to zero.
  2. Find two numbers that multiply to ( ac ) and add to ( b ).
  3. Rewrite and factor the equation.
• Example: Factor ( x^2 - 5x + 6 = 0 ) into ( (x - 2)(x - 3) = 0 ).

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Using the Quadratic Formula

• The Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
• Steps to use:
  1. Identify coefficients ( a, b, ) and ( c ).
  2. Calculate the discriminant ( D = b^2 - 4ac ).
     • If ( D > 0 ): two distinct real roots.
     • If ( D = 0 ): one real root.
     • If ( D < 0 ): no real roots.
• Example: For ( 3x^2 + 6x + 3 = 0 ), apply the formula.

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Graphing Quadratic Functions

• Key features of the graph:
  • Vertex: The highest or lowest point of the parabola.
  • Axis of symmetry: A vertical line that divides the parabola into two mirror images.
  • Intercepts: Points where the graph crosses the axes (x-intercepts and y-intercept).
• Example: Graphing ( y = x^2 - 4x + 3 ).

{The image of a graph showing the parabola of the equation ( y = x^2 - 4x + 3 ), highlighting the vertex, axis of symmetry, and x-intercepts.}

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Real-World Applications of Quadratic Equations

• Physics: Modeling the trajectory of thrown objects.
• Business: Optimization problems, like maximizing profit given constraints.
• Engineering: Calculating the dimensions of physical structures or materials.
• Example: A water fountain designs using quadratic equations to ensure a proper arc.

{The image of a water fountain demonstrating the arc of the water, shaped like a parabola, with a quadratic equation illustrated nearby.}

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Review and Practice Problems

• Recap key concepts:
  • Definition and standard form of quadratic equations.
  • Methods to solve (factoring, quadratic formula).
  • Graphing techniques and key features.
• Practice problems:
  1. Solve ( x^2 - 7x + 10 = 0 ) by factoring.
  2. Graph ( y = -2x^2 + 8x - 5 ) and identify the vertex and intercepts.
  3. Use the quadratic formula to find the roots of ( 2x^2 + 8x + 6 = 0 ).

{The image of a blackboard filled with practice problems about quadratic equations, with various solutions written down to illustrate solving methods.}