**Unformatted text preview: **t; 0.
4
Up if k > 0, down if k < 0
5
Actually, we could also take the standard form, f (x) = a(x − h)2 + k, expand it, and compare the coeﬃcients of
it and the general form to deduce the result. However, we will have another use for the completed square form of the
general form of a quadratic, so we’ll proceed with the conversion.
3 2.3 Quadratic Functions 143 we have derived the vertex formula for the general form as well. Note that the value a plays the
exact same role in both the standard and general equations of a quadratic function − it is the
coeﬃcient of x2 in each. No matter what the form, if a > 0, the parabola opens upwards; if a < 0,
the parabola opens downwards.
Now that we have the completed square form of the general form of a quadratic function, it is time
to remind ourselves of the quadratic formula. In a function context, it gives us a means to ﬁnd
the zeros of a quadratic function in general form.
Equation 2.5. The Quadratic Formula: If a, b, c are real numbers with a = 0, then the
solutions to ax2 + bx + c = 0 are
√
−b ± b2 − 4ac
x=
.
2a
Assuming the conditions of Equation 2.5, the solutions to ax2 + bx + c = 0 are precisely the zeros
of f (x) = ax2 + bx + c. We have shown an equivalent formula for f is
f (x) = a x + b
2a 2 + 4ac − b2
.
4a Hence, an equation equivalent to ax2 + bx + c = 0 is
a x+ b
2a 2 + 4ac − b2
= 0.
4a Solving gives a x+ b
2a 2 + 4ac − b2
4a a x+ 2 b
2a 1
b
a x+
a
2a
b
x+
2a
x+ =0
=− 2 4ac − b2...

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