Quadratics Flashcards
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, generally in the form ax² + bx + c = 0.
True or False: The standard form of a quadratic equation is y = ax² + bx + c.
True
What does the graph of a quadratic equation typically resemble?
The graph of a quadratic equation typically resembles a parabola.
What are the coefficients in the quadratic equation ax² + bx + c?
a is the coefficient of x², b is the coefficient of x, and c is the constant term.
Fill in the blank: The axis of symmetry of a parabola given by the equation y = ax² + bx + c is x = ___ .
-b/(2a)
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
What do the solutions of a quadratic equation represent?
The solutions of a quadratic equation represent the x-intercepts of the graph.
True or False: A quadratic equation can have zero, one, or two real solutions.
True
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a).
What does the discriminant in the quadratic formula indicate?
The discriminant (b² - 4ac) indicates the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; if negative, no real roots.
True or False: Completing the square is a method used to solve quadratic equations.
True
What is the first step in completing the square for the equation ax² + bx + c?
Divide all terms by a (if a ≠ 1).
What is the purpose of completing the square?
The purpose is to transform a quadratic equation into a perfect square trinomial.
Fill in the blank: When completing the square, the expression x² + bx is transformed into ___ .
(x + b/2)² - (b/2)²
What is the factored form of a quadratic equation?
The factored form is expressed as a(x - r₁)(x - r₂), where r₁ and r₂ are the roots.
What is the first step in factoring a quadratic equation?
Look for a common factor in all terms.
True or False: The roots of the quadratic equation ax² + bx + c can always be found by factoring.
False
What is the vertex form of a quadratic equation?
The vertex form is y = a(x - h)² + k, where (h, k) is the vertex.
Fill in the blank: The maximum or minimum value of a quadratic function occurs at the vertex, which is at x = ___ .
-b/(2a)
What does a positive ‘a’ value indicate about the parabola?
A positive ‘a’ value indicates that the parabola opens upwards.
What does a negative ‘a’ value indicate about the parabola?
A negative ‘a’ value indicates that the parabola opens downwards.
True or False: The y-intercept of a quadratic function can be found by evaluating the function at x = 0.
True
What is the y-intercept of the quadratic equation y = ax² + bx + c?
The y-intercept is c.
What is the relationship between the roots and the coefficients of the quadratic equation?
The sum of the roots is -b/a and the product of the roots is c/a.
