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What is a quadratic equation?
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A quadratic equation is a second-degree polynomial equation in a single variable x, with the highest exponent being 2, generally in the form ax² + bx + c = 0.
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What are the components of a quadratic equation?
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A quadratic equation has three main components: a (leading coefficient), b (linear coefficient), and c (constant term).
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What is the Quadratic Formula?
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The Quadratic Formula is x = (-b ± √(b² - 4ac)) / (2a), which is used to find the solutions of a quadratic equation.
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What is the discriminant in a quadratic equation?
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The discriminant is b² - 4ac and it determines the nature of the roots of the quadratic equation.
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How does the discriminant indicate the nature of the roots?
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If the discriminant is positive, there are two distinct real roots; if zero, there is one real root; if negative, there are two complex roots.
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What does 'solving by factoring' mean for quadratic equations?
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It involves expressing the quadratic equation as a product of its factors set to zero, then solving for x.
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How can completing the square solve a quadratic equation?
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This method involves creating a perfect square trinomial from the quadratic and solving for x.
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When is the quadratic formula used in solving quadratic equations?
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It is used when a quadratic equation cannot be easily factored.
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What is an example of a quadratic equation?
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An example is x² - 4x + 4 = 0.
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How do you convert a standard quadratic equation to vertex form?
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By completing the square and rearranging the equation into the form y = a(x - h)² + k.
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Why are quadratic equations important in real-world applications?
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They are used in physics, engineering, economics, and various sciences to model parabolic trajectories, optimize areas, and more.
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What graph is represented by a quadratic equation?
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A parabola, which may open upwards or downwards.
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What determines the direction of the parabola in a quadratic equation graph?
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The sign of the leading coefficient 'a'. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
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What are the vertex and axis of symmetry in a quadratic function?
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The vertex is the highest or lowest point of the parabola and the axis of symmetry is a vertical line through the vertex, x = -b/(2a).
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How is the axis of symmetry related to the quadratic formula?
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The axis of symmetry x = -b/(2a) is derived from the formula and represents the vertical line that divides the parabola into two symmetrical halves.
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