Flash Cards: Quadratic Equations
Quadratic Equations - Flash Cards
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What is a quadratic equation?
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A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a ≠ 0.
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What is the general form of a quadratic equation?
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The general form is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
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What does 'a' represent in a quadratic equation?
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'a' is the coefficient of x^2, determining the parabola's direction (upward if a > 0, downward if a < 0).
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What is the quadratic formula?
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The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a) used to find the roots of a quadratic equation.
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How can you determine the nature of the roots?
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The nature of the roots can be determined using the discriminant (D = b² - 4ac): If D > 0, two distinct real roots; D = 0, one real root; D < 0, no real roots.
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What is a perfect square trinomial?
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A perfect square trinomial is an expression of the form (a ± b)² = a² ± 2ab + b².
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Give an example of a quadratic equation from a real-life situation.
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The area of a hall can be modeled by the equation 2x² + x - 300 = 0, where 'x' represents its breadth in meters.
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What is meant by the roots of a quadratic equation?
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The roots of a quadratic equation are the values of x that satisfy the equation, i.e., where the graph intersects the x-axis.
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What does 'completing the square' mean?
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Completing the square is a method used to convert a quadratic equation into the form (x - p)² = q, facilitating the solution.
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What is the vertex of a parabola?
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The vertex is the highest or lowest point of the parabola, which occurs at x = -b/(2a) in a quadratic equation.
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Define the term 'discriminant'.
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The discriminant is the part of the quadratic formula under the square root, given by D = b² - 4ac, indicating the number of roots.
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What are the coefficients in the equation ax² + bx + c?
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'a', 'b', and 'c' are coefficients, with 'a' being the leading coefficient, 'b' the linear coefficient, and 'c' the constant term.
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What does it mean for roots to be 'real'?
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Real roots are solutions that are actual numbers; they exist when the discriminant is non-negative (D ≥ 0).
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How do you factor a quadratic equation?
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To factor a quadratic, express it as (px + q)(rx + s) such that the product equals ax² + bx + c.
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What is the significance of the Y-intercept in a quadratic equation?
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The Y-intercept is the point where the graph intersects the y-axis, found by evaluating the equation at x = 0, giving c.
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What role does the parabola play in a quadratic equation?
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The parabola graphically represents the quadratic equation, showing its symmetry, vertex, and roots.
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What is an 'irrational root'?
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An irrational root is a root that cannot be expressed as a simple fraction, typically occurring when D > 0 and is not a perfect square.
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Explain a common mistake when solving quadratic equations.
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A common mistake is miscalculating the discriminant or neglecting to check for extraneous roots after solving.
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What are linear and quadratic terms?
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Linear terms are of the form bx (first degree), while quadratic terms are of the form ax² (second degree).
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What is the connection between roots and x-intercepts?
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The roots of the quadratic equation correspond to the x-intercepts of its graph, where y = 0.
