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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Fourteen — Quadratic Equations and Inequalities
The Humongous Book of Algebra Problems
312
Solve the equation using the quadratic formula.
Reduce the fraction to lowest terms.
Squaring both sides of an equation might introduce false answers, so verify the
solutions by substituting them into the original equation.
Neither nor satisfies the equation, so the equation
has no real number solution.
Applying the Discriminant
What b
2
– 4ac tells you about an equation
14.29 The discriminant of the equation ax
2
+ bx + c = 0 is defined as the value
b
2
– 4ac. What conclusions can be drawn about the real roots of a quadratic
equation based on the sign of its discriminant?
According to the fundamental theorem of algebra, all quadratic equations have
two roots, because all quadratic equations have degree two. The discriminant is
calculated to predict how many (if any) of those roots will be real numbers.
Getting
to the last
steps from the
steps above them
is quite advanced,
so dont worry about
simplifying something
like
.
Instead, type the
expressions into a
calculator. The
decimal you get for
the left side wont
equal the decimal you
get for the right side,
and that’s enough
to show that the
equation is false.
Roots
is another
words for
solutions,” so the
books asking what
you can predict
about the real number
solutions of an equation.
Dont get these roots
confused with radicals;
square ROOTS
and the ROOTS
of an equation
mean totally
different
things.
Chapter Fourteen — Quadratic Equations and Inequalities
The Humongous Book of Algebra Problems
313
A positive discriminant indicates that the equation has two unique real roots.
In other words, there are two different, real number solutions to the quadratic.
The solutions may be rational (which means that the equation can be solved
by factoring) or irrational (which means that the equation must be solved by
completing the square or the quadratic equation), but both solutions are real
numbers.
A zero discriminant is an indication of a double root. The quadratic still has
two solutions, but the solutions are equal. For instance, the quadratic equation
(x + 5)(x + 5) = 0 has two roots: x = –5 and x = –5. Because a single real root is
repeated, x = –5 is described as a double root.
If a quadratic equation has a negative discriminant, there are no real number
solutions. That does not mean the equation has no solutions, but instead that
the solutions are complex numbers.
14.30 Given the equation 3x
2
+ 7x + 1 = 0, use the discriminant to predict how many
of the roots (if any) are real numbers, then calculate the roots to verify the
prediction made by the discriminant.
The equation has form ax
2
+ bx + c = 0, so substitute a = 3, b = 7, and c = 1 into
the discriminant formula.
According to Problem 14.29, the equation 3x
2
+ 7x + 1 = 0 has two positive
real roots because the discriminant is positive (37 > 0). Calculate the roots by
applying the quadratic formula.
The solution to the equation is or .
And they’ll
contain the
imaginary number
.
In case you
didnt recognize
it, the discriminant
b
2
– 4ac is the part
of the quadratic
formula that’s inside
the radical. When you
plug everything into the
quadratic formula,
theres no need to
calculate b
2
– 4ac
again—just plug in
the discriminant, 37.

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