quadratics notes

Quadratic equation standard form: ax^2 + bx + c = 0 This is the standard form for any quadratic equation, where a, b, and c are constants. It can be used to find the roots or zeros of the equation by using the quadratic formula.

Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a This formula is used to find the roots (x-intercepts) of a quadratic equation in standard form. It gives the two possible values for x when the equation is equal to zero.

Vertex form: y = a(x - h)^2 + k This form is useful for easily identifying the vertex of a quadratic equation, where (h, k) is the vertex point. It can be used to graph the parabola or find the maximum/minimum value.

Completing the square: x^2 + bx + c = a(x - h)^2 + k This method is used to convert a quadratic equation from standard form to vertex form. It involves adding and subtracting a constant term to make the left side a perfect square trinomial.

Discriminant: Δ = b^2 - 4ac The discriminant is used to determine the nature of the roots of a quadratic equation. It can help predict whether the equation has real or complex roots and how many distinct solutions it has.

Factored form: y = a(x - p)(x - q) This form is useful for quickly identifying the roots of a quadratic equation, where p and q are the x-intercepts. It can also be used to find the factors of the quadratic equation.

Axis of symmetry: x = -b / 2a This formula is used to find the vertical line that divides the parabola into two symmetrical halves. It can help in graphing the quadratic function and finding the vertex.

Parabola focus: F(h, k + 1/4a) The focus is a point that defines the geometric property of a parabola. It can be used to find the directrix and derive the equation of a parabola from its geometric definition.

Parabola directrix: y = k - 1/4a The directrix is a horizontal line that is equidistant from the focus and vertex of the parabola. It can be used in the geometric definition of a parabola and to find its equation from given points.

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