Quadratic Equations

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What is the standard form of a quadratic equation?
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The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
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What is the quadratic formula used for?
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The quadratic formula is used to find the solutions of a quadratic equation and is x = (-b ± √(b²-4ac)) / (2a).
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What is the discriminant in a quadratic equation?
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The discriminant is the part of the quadratic formula under the square root, b²-4ac. It determines the nature of roots.
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How does the discriminant determine the nature of solutions of a quadratic equation?
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If the discriminant > 0, there are two real and distinct solutions; if it = 0, there is one real double root; if < 0, there are two complex solutions.
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What are the roots of the quadratic equation?
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The roots of a quadratic equation are the values of x that satisfy the equation ax² + bx + c = 0.
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What methods can be used to solve quadratic equations besides the quadratic formula?
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Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
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What is the 'axis of symmetry' in a quadratic equation?
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The axis of symmetry of a quadratic function y = ax² + bx + c is the vertical line x = -b/(2a) that divides the parabola into two symmetric parts.
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Describe how factoring is used to solve quadratic equations.
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To solve by factoring, write the quadratic as a product of two binomials, set each binomial equal to zero, and solve for x.
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What is 'completing the square' and how is it used in solving quadratic equations?
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Completing the square involves rewriting the quadratic in the form (x - p)² = q, then solving for x by taking the square root of both sides.
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Explain how graphing can help find the solutions of a quadratic equation.
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Graphing the related quadratic function y = ax² + bx + c shows where the curve intersects the x-axis, which represents the solutions of the equation.
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What is the vertex of a parabola in the context of a quadratic function?
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The vertex is the highest or lowest point of a parabola, given by the point (h, k) where h = -b/(2a) and k is the y-value at x = h.
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How do you determine whether a quadratic function opens upwards or downwards?
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The sign of the coefficient a in the quadratic determines this; if a > 0, it opens upwards, and if a < 0, it opens downwards.
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What role does the leading coefficient play in stretching or compressing the graph of a quadratic?
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The leading coefficient a affects the width of the parabola; larger |a| values compress it, while smaller |a| values widen it.
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What are complex roots and when do they occur in quadratic equations?
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Complex roots occur when the discriminant is less than zero, leading to non-real solutions involving imaginary numbers.
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Can a quadratic equation have no real roots? Explain.
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Yes, if the discriminant is less than zero, the quadratic equation will have no real roots, only complex ones.
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