Completing the Square

Completing The Square

1. Make the quadratic equation on one side of the equal sign into a perfect square– Add to both sides to make the last term

correct

2. Take the square root of both sides

3. The numerical side gets a plus and minus

4. Simplify the variable side

Solve by taking the square root of each side. Round to the nearest tenth if necessary.

is a perfect square trinomial.

Original equation

Take the square root of each side.

Simplify.

Definition of absolute value

Subtract 3 from each side.

Use a calculator to evaluate each value of x.

Simplify.

or

Answer: The solution set is {–5.2, –0.8}.

Solve by taking the square root of each side. Round to the nearest tenth if necessary.

Answer: {–2.3, –5.7}

Find the value of c that makes a perfect square.

Complete the square.

Step 1 Find

Step 3 Add the result ofStep 2 to

Step 2 Square the result of Step 1.

Answer:

Notice that

Find the value of c that makes a perfect square.

Answer:

Perfect Square Process

The last term is one-half the middle term squared

e.g. x2 + 10x

The last term should be (½ * 10)2

= 25

Solve by completing the square.

Step 1 Isolate the x2 and x terms.

Original equation

Subtract 5 fromeach side.

Simplify.

Step 2 Complete the square and solve.

Take the square root of each side.

Since ,

add 81 to each side.

Factor

Add 9 to each side.

or

Simplify.

Check Substitute each value for x in the original equation.

Answer: The solution set is {1, 17}.

Solving a problem by completing the square

• Arrange terms as follows

x2 + bx = -c• Complete the square, adding the same constant

to both sides of the equation. (The last term is one-half the middle term squared)

• Square root of both sides• Solve for x, there can be up to two answers

Answer: {–2, 10}

Solve

Answer: {–5, 2}x x2 2 10 0 Solve

When a ≠ 1

• Divide every term by “a”, so that “a” does equal one.

• First step becomes

Arrange terms as follows

x2 + (b/a) x = (-c/a)

Homework

• 10-3 Completing the Square

Two Pages

First Column

Boating Suppose the rate of flow of an 80-foot-wide river is given by the equationwhere r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquim does not want to paddle his canoe against a current faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour?

Explore You know the function that relates distance from shore to the rate of the river current. You want to know how far away from the river bank he must paddle to avoid the current.

Plan Find the distance when Use completing the square to solve

Solve Equation for the current

Divide each side by –0.01.

Simplify.

Since

add 1600 to each side.

Factor

Take the square root of each side.

Add 40 to each side.

Simplify.

Use a calculator to evaluate each value of x.

or

Examine The solutions of the equation are about 7 ft and about 73 ft. The solutions are distances from one shore. Since the river is about 80 ft wide,

Answer: He must stay within about 7 feet of either bank.

Boating Suppose the rate of flow of a 6-foot-wide river is given by the equationwhere r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquim does not want to paddle his canoe against a current faster than 5 files per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour.

Answer: He must stay within 10 feet of either bank.