6. The vertex of the parabola can be found using the formula \( h = -\frac{b}{2a} \) and \( k = f(h) \) where \( f(h) = ah^2 + bh + c \).
7. The axis of symmetry of the parabola is the vertical line passing through the vertex, given by the equation \( x = h \).
8. The value of the discriminant, \( b^2 - 4ac \), can determine the nature of the solutions of the quadratic equation. If the discriminant is positive, there are two distinct real roots. If the discriminant is zero, there is one real root (a repeated root). If the discriminant is negative, there are no real roots (two complex roots).
9. The graph of a quadratic equation takes the form of a parabola. The direction of opening of the parabola (upward or downward) is determined by the sign of the coefficient \( a \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), the parabola opens downwards.
10. By analyzing the coefficients \( a \), \( b \), and \( c \) of the quadratic equation, you can identify features such as the vertex, axis of symmetry, direction of opening, x-intercepts (if any), and the discriminant to determine the nature of the roots.
Start with a general quadratic equation in the form of \( ax^2 + bx + c \).
2. Expand the equation by multiplying out the terms: \( a(x^2) + b(x) + c \).
3. Simplify the equation by combining like terms to get the quadratic form: \( ax^2 + bx + c \).
4. The quadratic form can also be expressed as \( a(x-h)^2 + k \), where \( h \) and \( k \) are the coordinates of the vertex of the parabola. This form can be obtained by completing the square on the original quadratic equation.
5. By deriving the quadratic form, you can gain insight into the vertex of the parabola, its direction of opening, and any transformations that have been applied to it.
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