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 Dependent Variable

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# The Square Root Property and Completing the Square

10.1 The Square Root Property and Completing the Square

An quadratic equation is written in standard form if it is written as ax2+bx+c =
0 with (a ≠ 0).

If x2 = k then OR .

Examples: Solve each equation.  Steps to Completing the Square for an equation of the form x2 + bx + c = 0:

1. But the terms with variables on one side and the constant on the other side.
2. Find .

3. Add the square to both sides.

4. Factor the perfect square trinomial on the LHS.

5. Solve the equation using the methods discussed earlier.

Let's practice step 2:

Find the constant that should be added to get a perfect square trinomial. Examples: Solve each equation by completing the square:  The solutions to ax2 + bx + c = 0(a ≠ 0) are given by Tips:

1. Make sure you put the equation in standard form FIRST!

Examples: Solve each equation using the quadratic formula.   What is wrong with the following "solution" of 5x2 - 5x + 1 = 0? Solve each equation: Solve each problem:

1. In 1.75 hr, Ken rows his boat 5 mi upriver and comes back. The speed of the
current is 3 mph. How fast does Ken row?

2. Two chefs are preparing a banquet. One chef could prepare the banquet in 2 hr
less time than the other. Together they complete the job in 5 hr. How long will it
take the faster chef working alone?   Summary of Ways to solve ax2 + bx + c = 0:

 Method Advantages Disadvantages Factoring This is usually the fastest method. Not all polynomials are factorable; some factorable polynomials are hard to factor. Square root property This is the simplest method for solving equations of the form (ax + b)2 = c. Few equations are given in this form. Completing the square This method can always be used, although most people prefer the quadratic formula. It requires more steps than other methods. Quadratic Formula This method can always be used. It is more difficult than factoring because of the square root, although calculators can simplify its use.

10.4 Formulas and Applications

Solve for the given variable: Word Problems:

Example 3 on pg. 592: Two cars left an intersection at the same time, one heading
due north, the other due west. Some time later, they are exactly 100 mi apart. The
car traveled?

#26 on pg. 596: A piece of sheet metal is 2 in. longer than it is wide. A square
piece 3 in. on a side is cut from each corner. The sides are then turned up to form an
uncovered box of volume 765 in.3. Find the dimensions of the original piece of metal.

#5 on pg. 594: A ball is propelled vertically upward from the ground. Its distance
in feet from the ground at t seconds is s(t) = -16t2 + 64t. At what times will the
ball be 32 ft from the ground?

The graph of a quadratic equation is called a parabola.

1. The graph of f(x) = a(x - h)2 + k, a ≠ 0 is a parabola with vertex (h, k) and
vertical line x = h as axis.

2. The graph opens up if a is positive and down if a is negative.

3. The graph is winder than that of f(x) = x2 if 0 < lal < 1. The graph is narrower
than that of f(x) = x2 if lal> 1.

Let's practice determining the vertex of a parabola and whether it opens upward or
downward. Graph the following. Give the vertex, axis, domain and range.   