Yes and this will happen when the discriminant of a quadratic equation is less than zero meaning it has no real roots.
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No. While the sum of two qudratics cannot have a power greater than two, it could have a power of 2, 1 or 0. x2 + 1 is one quadratic (2-x)(2+x) is another quadratic. Their sum is 1, a constant (power = 0).
No, but they are symmetric with respect to a line parallel to the y-axis - which could be the y-axis itself.
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As one example, the quadratic equation can be used to model many different phenomena - if you were to measure the height of a baseball as it was thrown straight up in the air and pulled down again, and plot the height with respect to time, it would look like a quadratic equation. Computer modelling is a large field with many applications in physics, meterology, and even social sciences.
Symmetry