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In this session you will learn to•Graph quadratic functions,•Solve quadratic equations.•Graph exponential functions•Solve problems involving exponential growth and decay.•Recognize and extend geometric sequences.
In this lesson you will learn to•Solve quadratic equations by finding the square root.•Solve quadratic equations by completing the square.
If the trinomial on the left side of the equation is a binomial square,
64129 2 xx
If the trinomial on the left side of the equation is a binomial square,
factor the trinomial,
64129 2 xx
6232
x
If the trinomial on the left side of the equation is a binomial square,
factor the trinomial,
64129 2 xx
6232
x
then take the square root of both sides. 6232
x
If the trinomial on the left side of the equation is a binomial square,
factor the trinomial,
64129 2 xx
6232
x
then take the square root of both sides. 6232
x
Simplify. 623x
If the trinomial on the left side of the equation is a binomial square,
factor the trinomial,
64129 2 xx
6232
x
then take the square root of both sides. 6232
x
Simplify. 623x
Solve for the unknown variable byadding 2
623x
If the trinomial on the left side of the equation is a binomial square,
factor the trinomial,
64129 2 xx
6232
x
then take the square root of both sides. 6232
x
Simplify. 623x
Solve for the unknown variable byadding 2
623x
and dividing by 3.
362x
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
•Remove the third term (7) of the trinomial by subtracting. 1___122 xx
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
•Remove the third term (7) of the trinomial by subtracting. 1___122 xx
•Complete the square by adding the square of half the coefficient of the second term.
2
2122
2122 112xx
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
•Remove the third term (7) of the trinomial by subtracting. 1___122 xx
•Complete the square by adding the square of half the coefficient of the second term.
2
2122
2122 112xx
•Simplify.3536122 xx
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
•Remove the third term (7) of the trinomial by subtracting. 1___122 xx
•Complete the square by adding the square of half the coefficient of the second term.
2
2122
2122 112xx
•Simplify.3536122 xx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
3562
x
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
•Remove the third term (7) of the trinomial by subtracting. 1___122 xx
•Complete the square by adding the square of half the coefficient of the second term.
2
2122
2122 112xx
•Simplify.3536122 xx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
3562
x
•Take the square root of both sides. 356x
If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
Solve 67122 xx by completing the square.
•Remove the third term (7) of the trinomial by subtracting. 1___122 xx
•Complete the square by adding the square of half the coefficient of the second term.
2
2122
2122 112xx
•Simplify.3536122 xx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
3562
x
•Take the square root of both sides. 356x
•Solve for the unknown variable by adding or subtracting.
356x
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
•The first step is to divide the entire equation by the leading coefficient. Divide by 2.
36272
26
27
2122
22
xx
xx
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
212
272 _63_6 xxxx
•The first step is to divide the entire equation by the leading coefficient. Divide by 2.
36272
26
27
2122
22
xx
xx
•Subtract .27
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
212
272 _63_6 xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
26
212
262 6xx
•Simplify.2
1722
2122 96336 xxxx
•The first step is to divide the entire equation by the leading coefficient. Divide by 2.
36272
26
27
2122
22
xx
xx
•Subtract .27
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
212
272 _63_6 xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
26
212
262 6xx
•Simplify.2
1722
2122 96336 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left. 2
1723x
•The first step is to divide the entire equation by the leading coefficient. Divide by 2.
36272
26
27
2122
22
xx
xx
•Subtract .27
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
212
272 _63_6 xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
26
212
262 6xx
•Simplify.2
1722
2122 96336 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left. 2
1723x
•Take the square root of both sides.2
173x
•The first step is to divide the entire equation by the leading coefficient. Divide by 2.
36272
26
27
2122
22
xx
xx
•Subtract .27
If there is a leading coefficient, there is one additional step at the beginning.
Solve 67122 2 xx by completing the square.
212
272 _63_6 xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
26
212
262 6xx
•Simplify.2
1722
2122 96336 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left. 2
1723x
•Take the square root of both sides.2
173x
•Solve for the unknown variable by adding or subtracting. 2
173x
•The first step is to divide the entire equation by the leading coefficient. Divide by 2.
36272
26
27
2122
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
•Divide by 2.
027
252
20
27
252
22
xx
xx
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
25
21
27
2
25
21
252 xx
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
25
21
27
2
25
21
252 xx
•Simplify. 1681
1625
27
1625
2522
45
27
2
45
252 xxxx
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
25
21
27
2
25
21
252 xx
•Simplify. 1681
1625
27
1625
2522
45
27
2
45
252 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
1681
2
45x
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
25
21
27
2
25
21
252 xx
•Simplify. 1681
1625
27
1625
2522
45
27
2
45
252 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
1681
2
45x
•Take the square root of both sides.29
481
45x
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
25
21
27
2
25
21
252 xx
•Simplify. 1681
1625
27
1625
2522
45
27
2
45
252 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
1681
2
45x
•Take the square root of both sides.29
481
45x
•Solve for the unknown variable by adding or subtracting.
44
414
49
45 ,x
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 0752 2 xx by completing the square.
27
252
27
252 _0_ xxxx
•Complete the square by adding the square of half the coefficient of the second term.
2
25
21
27
2
25
21
252 xx
•Simplify. 1681
1625
27
1625
2522
45
27
2
45
252 xxxx
•The trinomial on the left is now a perfect square. Factor the trinomial on the left.
1681
2
45x
•Take the square root of both sides.29
481
45x
•Solve for the unknown variable by adding or subtracting.
44
414
49
45 ,x
•Divide by 2.
027
252
20
27
252
22
xx
xx
•Subtract .27
Reduce: 1,27x
