W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

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Chabot Mathematics
7.6 2RadRadical Eqns
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 7.6 ? Radical Equations
• Any QUESTIONS About HomeWork
• 7.6 ? HW-29

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Radical Equations

• A Radical Equation is an equation in which at
  least one variable appears in a radicand.
• Some Examples

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Solve Eqns with 2 Rad. Terms

1. Isolate one of the radical terms.
2. Use the Exponent Power Rule
3. If a radical remains, perform steps (1) and (2)
   again.
4. Solve the resulting equation.
5. Check the possible solutions in the original
   equation.

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Example ? Solve

• SOLUTION

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Example ? Solve

• SOLUTION

• Check 6 by Inspection ? 3-21 ?
• Thus The number 6 checks and it IS the solution

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Example ? Solve

• SOLN

One radical is isolated. We square both sides.
Square both sides.
Factoring
Using the principle of zero products
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Example ? Solve

• Check x 3 x 11

?
?

• The numbers 3 and 11 check and are then
  confirmed as solutions.

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Example ? Solve
Start by isolating one radical on one side
of the equation by subtracting from each
side. Then square both sides.
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Example ? Solve
This equation still contains a radical, so square
both sides again. Before doing this, isolate the
radical term on the right.
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Example ? Solve
This equation still contains a radical, so square
both sides again.
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Example ? Solve
Now finish solving the equation.
x 3 or x 2
Finally CHECK for Extraneous Solutions
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Example ? Solve
Check each potential solution, 3 and 2, in the
original equation.
If x 3, then
If x 2, then
?
?
5 5
5 5
The solution set is -3, 2 .
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The Principle of Square Roots

• Recall the definition of the PRINCIPAL Square
  Root
• For any NONnegative real number n, If x2 n,
  then

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Recall The Pythagorean Theorem

• In any right triangle, if a and b are the lengths
  of the legs and c is the length of the
  hypotenuse, then
• a2 b2 c2

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Example ? Pythagorus

• How long is a guy wire if it reaches from the top
  of a 14 ft pole to a point on the ground 8 ft
  from the pole?

• SOLUTION

• We now use the principle of square roots. Since
  d represents a length, it follows that d is the
  positive square root of 260

Diagram
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Isosceles Right Triangle

• When both legs of a right triangle are the same
  size, we call the triangle an isosceles right
  triangle. If one leg of an isosceles right
  triangle has length a then

c
a
a
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Lengths for Isosceles Rt Triangles

• The length of the hypotenuse in an isosceles
  right triangle is the length of a leg times

45o
a
45o
a
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Example ? Isosceles Rt. Tri.

• The hypotenuse of an isosceles right triangle is
  8 ft long. Find the length of a leg. Give an
  exact answer and an approximation to three
  decimal places.

• SOLUTION

( after Rationalizing Divisor).
Exact answer Approximation
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30-60-90 Triangle

• A second special triangle is known as a
  30-60-90 right triangle, so named because of
  the measures of its angles

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Lengths for 30/60/90 Rt Triangles

• The length of the longer leg in a 30/60/90 right
  triangle is the length of the shorter leg times
  The hypotenuse is twice as long as the
  shorter leg.

30o
2a
60o
a
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Example ? 30-60-90 Triangle

• The shorter leg of a 30/60/90 right triangle
  measures 12 in. Find the lengths of the other
  sides. Give exact answers and, where
  appropriate, an approximation to three decimal
  places.

• SOLUTION

• The hypotenuse is twice as long as the shorter
  leg, so we have

c 2a
2(12) 24 in.
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Example ? 30-60-90 Triangle

• SOLUTION
• The length of the longer leg is the length of
  the shorter leg times This yields

Exact answer Approximation
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WhiteBoard Work

• Problems From 7.6 Exercise Set
• 24, 34, 38, 48, 60

• Astronomical Unit Sun?Earth Distance
  149 598 000 km

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All Done for Today
The SolarStar System
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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

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Graph y x

• Make T-table

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