Questions on Algebra: Quadratic Equation answered by real tutors!

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How many solutions exist for a quadratic equation?
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The number of roots of a polynomial equation is equal to the degree of the polynomial (the exponent of the leading term).
Quadratic equations are of degree 2, .
They have two (2) roots.
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How do we detemine whether the solutions are real or complex?
Use the discriminant.
For the general quadratic equation,

the discriminant is

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If then you have two distinct real roots.
Example:


2 real roots, x=2,5 .
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If , you have a double root, one real root occurring twice


2 real roots, x=1,1.
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If , you have two complex roots, that are complex conjugates.


2 complex roots, x=i,-i.

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In general, the number of solutions for a polynomial is equal to the degree of the polynomial.
A quadratic equation is a polynomial of degree 2 so it would have 2 solutions. The type of solutions a quadratic equation can be determined by examining the discriminant: which is taken from the quadratic formula:
If the discriminant is negative, there are no real solutions/roots. This makes sense when you realize that a negative discriminant (the square root of a negative quantity) will yield complex solutions.
If the discriminant is zero, there is one real solution/root, sometimes referred to as a double root because you get two real solutions that are identical.
If the discriminant is positive, there are two real solutions/roots.
It is helpful to look at the graphs of quadratic equations with the above type of solutions/roots:

Green graph: Discriminant is negative, no real roots.
Red graph: Discriminant is zero, one double root.
Blue graph: Discriminant is positive, two real root.
As you can see, the roots or solutions to these equations are the x-values where the curves (parabolas) intersect the x-axis.