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What is the general form of a quadratic equation?
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The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
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How can you find the roots of a quadratic equation using the quadratic formula?
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The roots can be found using the formula: x = (-b ± √(b^2 - 4ac)) / (2a).
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What are the types of roots a quadratic equation can have?
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The roots can be real and distinct, real and equal, or complex (non-real), depending on the value of the discriminant.
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What is the discriminant in a quadratic equation, and what does it determine?
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The discriminant is b^2 - 4ac. It determines the nature of the roots: if >0, roots are real and distinct; if =0, roots are real and equal; if <0, roots are complex.
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How does factoring help solve quadratic equations?
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If a quadratic can be factored into (px + q)(rx + s) = 0, the roots are x = -q/p and x = -s/r.
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What is completing the square, and how is it used to solve quadratic equations?
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Completing the square involves rewriting ax^2 + bx + c in the form (x – h)^2 = k to solve for x.
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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 graphing a quadratic equation help find its solutions?
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The solutions (or roots) of the equation correspond to the x-intercepts of its graph.
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What role does the leading coefficient 'a' play in the graph of a quadratic equation?
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The leading coefficient 'a' determines the direction of the parabola opening; if a > 0, it opens upwards, and if a < 0, it opens downwards.
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How do you determine the axis of symmetry for a quadratic equation in standard form?
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The axis of symmetry is x = -b/(2a).
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How is the quadratic formula derived?
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The quadratic formula is derived by completing the square on the general form ax^2 + bx + c = 0.
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Can a quadratic equation be solved through graphing?
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Yes, a quadratic equation can be solved by finding the x-intercepts of its graph.
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What is the sum of the roots in a quadratic equation?
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The sum of the roots, according to Vieta's formulas, is -b/a.
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What is the product of the roots in a quadratic equation?
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The product of the roots, according to Vieta's formulas, is c/a.
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Why can't a be zero in a quadratic equation?
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If a were zero, the equation would not be quadratic, as ax^2 would be eliminated, reducing it to a linear equation.
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