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## Quadratic EquationsI. Finding Roots of a Quadratic Equation
2x The values that satisfy a quadratic (or any polynomial
equation) are called roots. There are 3 primary methods for finding roots to a quadratic. Here are examples and comments on each. A. Factoring Consider the equation 2x ^{2} + x − 6 = 0. When expressed as a
polynomial,its roots are not easily apparent. Notice what happens if we rewrite this expression in factored form: The roots now become clear:
or x = −2. When solving a quadratic
What if you attempted the same problem using the following method?
Every quadratic equation has 2 roots. Dividing by x
removes the root x = 0.
Here's one last example of how factoring finds roots.
Answers
Any help you need with simplifying radicals can be found
in
For quadratics with no middle term (when b = 0), the simplest approach is to take square roots of both sides of the equation. Example: Solve 3x^{2} − 4 = 0.Solution:
Common error: Don't forget the negative root.
It will help if you keep this example in mind when looking
at Section III of for v (Kinetic energy) Answers
You should now be able to solve quadratic equations using any of the three methods shown: factoring, quadratic formula, or taking roots. Here is a summary of what has been covered. 1) For ax ^{2}+c = 0, isolate x^{2} and square root both sides.
Don't forget thenegative root. Otherwise... 2) Put into the form ax ^{2} + b x + c = 0. This may require removing paren-theses or clearing fractions. Dividing out a constant is helpful but not necessary. 3) Find roots by factoring or the *quadratic formula. If b ^{2} − 4ac <
0, theequation has no real roots. 4) Check solutions, especially if original equation is fractional. *Don't overuse the quadratic formula. Factoring is an important skill to maintain so use it at every opportunity.
Section Ic) demonstrated how quadratics in the form (x ( )) ^{2} = k aresolved. Illustration: (How is this related to completing the square? By expanding (x − 2) ^{2}
andsetting equal to 0, This would seem to indicate that any quadratic can be changed into (x ±( )) ^{2} = k form
(andthen solved). Such a process is called completing the square. A. Perfect Square TrinomialsCompleting the square requires a thorough understanding of how trinomials of the form always factor into (Review Topic 4).
Trinomials such as these are referred to as Perfect Square
Trinomials (PST). Answers
Why are these the roots of 2x if
Answers Solve for x. Hint: Find LCD and clear fractions. Since (2x + 5) is a repeated factor,
is a repeated or double root. Thus x = 0 or
appear to be roots. Since division by 0 is
not Find all real solutions. for v (Kinetic energy)
Since b f) After clearing fractions, 5x
Return to Review Topic Find the term needed to make a PST, then express in factored form.
Return to Review Topic Solve by completing the square.
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