Quadratic Equations Mastery

This cheat sheet covers key concepts and problem-solving techniques for quadratic equations, including completing the square, using the quadratic formula, and analyzing solutions via the discriminant. It also includes applications in real-world scenarios.

Core Principles

Perfect square trinomials are expressions that can be factored into the square of a binomial.
Completing the square is a method to rewrite a quadratic expression into a perfect square trinomial plus a constant.
The quadratic formula provides solutions for any quadratic equation.
The discriminant (b² - 4ac) determines the number and type of solutions for a quadratic equation.
Quadratic equations model parabolic motion, useful in physics and optimization problems.

Action Steps

To find 'c' in a perfect square trinomial $x^2 + bx + c$, calculate $(b/2)^2$.
To check if an expression is a perfect square trinomial, see if it fits the form $(ax )^2$ or $(ax - b)^2$.
To rewrite a perfect square trinomial as a binomial squared, take the square root of the first and last terms, using the sign of the middle term.
To solve by completing the square: Move the constant term to the right side, add $(b/2)^2$ to both sides, factor the left side, and take the square root of both sides.
To determine the number of solutions using the discriminant: If $\Delta > 0$, two real solutions. If $\Delta = 0$, one real solution. If $\Delta < 0$, no real solutions.
To find the maximum or minimum of a quadratic function $h(t) = at^2 + bt + c$, use $t = -b / (2a)$ to find the time, then substitute back into the function to find the maximum height.

Formulas

Completing the square: For $x^2 + bx$, add $(b/2)^2$ to make it a perfect square trinomial: $(x + b/2)^2$
Quadratic Formula: For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $\Delta = b^2 - 4ac$
Vertex of a parabola (for maximum/minimum): $t = -b / (2a)$

Key Terms

Perfect Square Trinomial: A trinomial that can be factored into the square of a binomial, e.g., $x^2 + 6x + 9 = (x+3)^2$.
Completing the Square: A method to solve quadratic equations by adding a constant to create a perfect square trinomial.
Quadratic Formula: A formula used to find the roots of a quadratic equation.
Discriminant: The part of the quadratic formula under the square root ($b^2 - 4ac$), which indicates the nature of the roots.
Parabola: The U-shaped graph of a quadratic function.
Vertex: The highest or lowest point on a parabola, representing maximum or minimum values.

Real World Examples

Maximizing rectangular area with fixed fencing.: Using the vertex formula to find dimensions that yield the largest area for a given perimeter.
Projectile motion (e.g., bunny population, water balloon launch).: Using quadratic functions to model the path of objects, finding maximum height, time to reach maximum height, and time to hit the ground.

Quiz

What is the value of 'c' that makes $x^2 + 10x + c$ a perfect square trinomial?: 25
How many real solutions does $2x^2 + 4x + 3 = 0$ have?: None
For the equation $h(t) = -16t^2 + 58t + 4$, what is the time to reach maximum height?: 1.81 seconds