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2.10 Special Quadratic Equations Theorem of Pythagoras from Basic Algebra: One Step at a Time © 2002 p. 193 - 200 Dr. Robert J. Rapalje Seminole State College of Florida Sanford, FL 32773
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
Before introducing the Theorem of Pythagoras, we begin with some perfect square equations. Perfect square equations (see the first example and the exercises that follow) can be solved by taking the square root of both sides of the equation. This is called the square root property of equations. When you use this property, you must include a "±" (that is, "+" or "-") in order to obtain both solutions of the equation.
EXAMPLE 1. Solve the equation x2 = 16. Solution: The solution is essentially to answer the question, "What number can be squared (multiplied times itself!) in order to get 16. There are actually two answers: x = 4 and also x = -4. This answer may be also written as x = ± 4.
EXAMPLE 2. Solve the equation x2 = 5. Solution: Unlike the first example, there is no whole
number or integer that you can square in order to get 5. It is possible,
however, to take the square root of both sides and write
EXAMPLE 3. Solve the equation x2 + 122 = 152. Solution: x2 + 122 = 152 You know that 122 = 144, and 152 = 225 (or use calculator!) x2 + 144 = 225 Subtract 144 from each side. - 144 - 144 x2 = 81 Because of the x2 , you have to have two answers: "±". x = ± 9.
EXAMPLE 4. Solve the equation x2 + 102 = 152. Solution: x2 + 102 = 152 You know that 102 = 100, and 152 = 225 (or use calculator!) x2 + 100 = 225 Subtract 100 from each side. - 100 - 100 x2 = 125 Because there is no "even" answer, use the square root.
x » ± 11.180. EXERCISES. Solve the following perfect square equations. In some, a calculator is needed! 1. x2 = 9 2. x2 = 25 3. x2 = 49 4. x2 = 169 x = ±_____ x = _____ x = _____ x = _____
5. x2 = 81 6. x2 = 36 7. x2 = 144 8. x2 = 121 x = _____ x = _____ x = _____ x = _____
9. x2 = 6 10. x2 = 30 11. x2 = 200 12. x2 = 120 x = _____ x = _____ x = _____ x = _____
13. 62 + 82 = x2 14. x2 + 52 = 132 15. 152 + x2 = 172 16. x2 = 32 + 42
17. 52 + 62 = x2 18. x2 + 102 = 132 19. 132 + x2 = 172 20. x2 = 122 + 92
21. 402 + 422 = x2 22. x2 + 242 = 252 23. 702 + x2 = 742 24. x2 = 132 + 842
The Theorem of Pythagoras is one of the most important formulas in all of mathematics. Although this theorem was known to the Babylonians 1000 years earlier, the credit for the first proof was given to the Greek mathematician Pythagoras, 6th century B.C. The Theorem of Pythagoras deals specifically with right triangles. In a right triangle, the two sides that are mutually perpendicular are called legs, and the third side, always opposite the right angle, and always the longest side, is called the hypotenuse of the triangle. According to the Theorem of Pythagoras, if "a" and "b" are legs, and "c" is the hypotenuse, then a2 + b2 = c2.
ANSWERS 1.10 p. 140-145:
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Using a calculator (see Section 1.04) you can give the decimal
approximation which is x »
± 2.236 (round off to nearest thousandth. Note: the wavy equal
sign "»" means "approximately
equal.")
Don’t forget the ±, and round off to
nearest thousandth.
