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The solutions to a quadratic equation can be found directly from the
quadratic formula.
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The equation
ax2 + bx +
c = 0
has solutions
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The advantage of using the formula is that it always works. The disadvantage
is that it can be more time-consuming than some of the methods previously
discussed. As a general rule you should look at a quadratic and see if it can
be solved by taking square roots; if not, then if it can be easily factored;
and finally use the quadratic formula if there is no easier way.
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Notice the plus-or-minus symbol (±) in
the formula. This is how you get the two different solutions—one using the plus
sign, and one with the minus.
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Make sure the equation is
written in standard form before reading off a, b, and c.
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Most importantly, make sure
the quadratic expression is equal to zero.
The formula requires you to take the square root of the expression b2
– 4ac, which is called the discriminant because it determines the
nature of the solutions. For example, you can’t take the square root of a
negative number, so if the discriminant is negative then there are no
solutions.
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If b2 – 4ac > 0
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There are two distinct real roots
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If b2 – 4ac = 0
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There is one real root
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If b2 – 4ac < 0
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There are no real roots
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The quadratic formula can be derived by using the technique of completing
the square on the general quadratic formula:
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Given:
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Divide through by a:
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Move the constant term to the right
side:
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Add the square of one-half the
coefficient of x to both sides:
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Factor the left side (which is now a
perfect square), and rearrange the right side:
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Get the right side over a common
denominator:
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Take the square root of both sides
(remembering to use plus-or-minus):
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Solve for x:
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