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What is the standard form of a quadratic equation?
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The standard form is ax^2 + bx + c = 0, where a, b, and c are constants and x represents an unknown variable.
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How do you determine if a quadratic equation has real roots using the discriminant?
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Calculate the discriminant using the formula b^2 - 4ac. If it is positive, there are two real roots; if zero, there is one real root; if negative, there are no real roots.
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What is the quadratic formula used to solve quadratic equations?
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The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
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What does it mean for a quadratic equation to have complex roots?
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Complex roots occur when the discriminant (b^2 - 4ac) is negative, indicating that the solutions involve imaginary numbers.
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How can factoring be used to solve quadratic equations?
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If a quadratic can be factored into (mx + n)(px + q) = 0, then the roots are found by setting each factor equal to zero and solving for x.
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What role do ‘a’, ‘b’, and ‘c’ play in a quadratic equation?
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In ax^2 + bx + c = 0, 'a' determines the parabola's direction and opening width, 'b' affects the position along the x-axis, and 'c' is the y-intercept.
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What is the vertex form of a quadratic equation?
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The vertex form is y = a(x – h)^2 + k, where (h,k) is the vertex of the parabola.
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How can you find the vertex of a quadratic function given its standard form?
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Use the formula h = -b/(2a) to find the x-coordinate, and substitute it back into the equation to find the y-coordinate.
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When is a quadratic considered a perfect square trinomial?
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When it can be expressed in the form (ax)^2 + 2abx + b^2 = 0, representing (ax + b)^2.
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How does completing the square help in solving quadratic equations?
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It transforms a quadratic equation into a perfect square trinomial, making it easier to solve by taking the square root of both sides.
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What is the relationship between the coefficients and the sum and product of the roots of a quadratic equation?
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For ax^2 + bx + c = 0, the sum of the roots is -b/a and the product is c/a.
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What is the graph of a quadratic function called?
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The graph of a quadratic function is called a parabola.
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What are the steps to graph a quadratic equation?
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Identify the vertex, axis of symmetry, x-intercepts, y-intercept, and plot these points to sketch the parabola.
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How does the value of 'a' affect the direction of a parabola?
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If 'a' is positive, the parabola opens upwards; if negative, it opens downwards.
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What is axis of symmetry in a quadratic function?
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The axis of symmetry is a vertical line x = -b/(2a) that divides the parabola into two symmetrical halves.
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