4.2 Zeros by Completing the Square
Overview of Quadratics
Focus on solving quadratics by completing the square.
Quadratics are parabolic graphs that are fundamental in algebra.
Characteristics of Parabolas
A standard parabola has a specific shape and orientation.
The downward shift (e.g., -5) affects the graph's vertical position.
Y-Intercept
To find the y-intercept, substitute x = 0 into the equation.
Graphically, this is where the parabola intersects the y-axis.
Patterns
For each unit moved along the x-axis, the y-value changes according to the square of the x-value.
Example: over 1 unit, up 1 squared; over 2 units, up 4; over 3 units, up 9.
X-Intercepts (Roots)
Quadratic equations can have multiple x-intercepts.
Example of x^2 - 4 which has x-intercepts at -2 and 2.
X-intercepts may be fractions or decimals if they don't align neatly on a grid.
Symmetry of Parabolas
Parabolas are symmetrical about their axis, allowing for predictions about the location of roots.
If one root is positive, the opposite will be negative.
Solving Quadratics
Factoring Method: Recognize patterns to factor quickly.
Example: For x^2 - 12x + 36, you can recognize it as (x - 6)^2.
Double Roots
A double root occurs when both factors are identical (e.g., x = 6).
Graphically, the parabola touches the x-axis at one point.
The y-intercept indicates the value of the function when x = 0.
Completing the Square
Essential for finding zeros of quadratics that do not factor easily.
Steps to complete the square:
Isolate the quadratic on one side.
Add to both sides to achieve a perfect square trinomial.
Factor and solve using the square root method.
Example Steps
Convert x^2 + b*x + c to (x + p)^2 equation.
Identify half of the middle coefficient, square it, add to both sides.
Finding Roots
Always include a plus/minus sign when taking square roots to account for both solutions.
Visual Analysis
Understand how transformations affect the graph:
Shifts dictate where the vertex and intercepts lie.
Vertical shifts change the height but not the width or orientation.
Recognizing when a quadratic does not intersect the x-axis indicates that there are no real solutions.
Summary of Solution Techniques
Factoring: Fastest method if possible.
Completing the Square: Reliable method especially when factoring isn’t straightforward.
Quadratic Formula: Useful for any quadratic form, provides a consistent way to find x-intercepts.
Practice Problems
Identify x-intercepts and y-intercepts for various parabolic equations.
Complete the square to find roots for more complex quadratics.