Solving Quadratic Equations by Extracting Square Roots

You will learn how to solve quadratic equations by extracting square roots.

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1. Grade 9 –Mathematics
Quarter I
SOLVINGQUADRATIC EQUATIONS BY
EXTRACTING SQUARE ROOTS

2. Objectives:
1.familiarizenumbersthatare
perfectsquares;and
2.solvequadraticequationsby
extractingsquareroots.

4. QuadraticEquationsthatcanbewritteninthe form ??????
�
=??????can be solved
by applying the following properties:
1.If ??????>0,then ??????
�
=??????has two real solutions or roots: ??????=±??????.
2. If ??????=0,then ??????
�
=??????has two real solutions or roots: ??????=0
3. If ??????<0,then ??????
&#3627409360;
=??????has no solutions or roots:
Speaking Mathematically
??????=±36isreadas‘??????isequaltothepositiveornegativesquarerootof36’
??????−5
2
isreadas“thesquarerootofthesquareofthequantity??????&#3627408526;??????&#3627408527;????????????&#3627409363;".

5. ??????
2
=49
??????
2
=±49
??????=±7
How to extract square roots?

6. ??????
2
=169
??????
2
=±169
??????=±13
How to extract square roots?

7. 8=4∙232=16∙254=9∙675=25∙3
12=4∙340=4∙1056=4∙1476=4∙19
18=9∙244=4∙1154=9∙680=16∙5
20=4∙545=9∙560=4∙1590=9∙10
24=4∙648=16∙363=9∙796=16∙6
27=9∙350=25∙268=4∙1798=49∙2
28=4∙752=4∙1372=36∙299=9∙11

8. 8=4∙2 54=9∙6
12=4∙3 56=4∙14
18=9∙2 54=9∙6
20=4∙5 60=4∙15
24=4∙6 63=9∙7
27=9∙3 68=4∙17
28=4∙7 72=36∙2
32=16∙2 75=25∙3
40=4∙10 76=4∙19
44=4∙11 80=16∙5
45=9∙5 90=9∙10
48=16∙3 96=16∙6
50=25∙2 98=49∙2
52=4∙13 99=9∙11
??????
2
=75
Get the square root of both sides
??????
2
=75
Factor the perfect squares??????=±25∙3
Get the square root of the
perfect square.??????=±253
??????=±53

9. &#3627409360;??????−&#3627409363;
&#3627409360;
=&#3627409361;&#3627409360;
Divide both sides by 2
??????−&#3627409363;
&#3627409360;
=&#3627409359;&#3627409364;
Get the square root of both sides??????−&#3627409363;
&#3627409360;
=&#3627409359;&#3627409364;
??????−&#3627409363;=±&#3627409362;
??????−&#3627409363;=&#3627409362;
??????=&#3627409362;+&#3627409363;
??????=&#3627409367;
??????−&#3627409363;=−&#3627409362;
??????=−&#3627409362;+&#3627409363;
??????=&#3627409359;
Find the solutions or roots.
&#3627409360;??????−&#3627409363;
&#3627409360;
=&#3627409361;&#3627409360;
&#3627409360;

10. &#3627409361;&#3627409362;??????−&#3627409359;
&#3627409360;
−&#3627409359;=&#3627409359;&#3627409359;
Divide both sides by 3 &#3627409361;&#3627409362;??????−&#3627409359;
&#3627409360;
=&#3627409359;&#3627409360;
Get the square root of both sides
&#3627409362;??????−&#3627409359;
&#3627409360;
=&#3627409362;
&#3627409362;??????−&#3627409359;=±&#3627409360;
&#3627409362;??????−&#3627409359;=&#3627409360;
4??????=&#3627409361;
??????=
&#3627409361;
&#3627409362;
&#3627409362;??????−&#3627409359;=−&#3627409360;
4??????=−&#3627409359;
??????=
−&#3627409359;
&#3627409362;
Find the solutions or roots.
&#3627409362;??????−&#3627409359;
&#3627409360;
=&#3627409362;
&#3627409361;&#3627409362;??????−&#3627409359;
&#3627409360;
=&#3627409359;&#3627409360;
&#3627409361;

11. &#3627409360;??????−&#3627409361;
&#3627409360;
=&#3627409359;&#3627409366;
Get the square root of both sides
2??????−3
2
=18
2??????−3
2
=±9∙2
2??????−3=±92
2??????−3=±32
2??????−3=±32
&#3627409360;??????−&#3627409361;=&#3627409361;&#3627409360;
&#3627409360;??????=&#3627409361;+&#3627409361;&#3627409360;
??????=
&#3627409361;+&#3627409361;&#3627409360;
&#3627409360;
&#3627409360;??????−&#3627409361;=−&#3627409361;&#3627409360;
&#3627409360;??????=&#3627409361;−&#3627409361;&#3627409360;
??????=
&#3627409361;−&#3627409361;&#3627409360;
&#3627409360;

12. &#3627409360;&#3627409363;??????+&#3627409360;
&#3627409360;
=&#3627409364;&#3627409362;
Get the square root of both sides
5??????+2
2
=32
5??????+2
2
=±16∙2
5??????+2=±162
5??????+2=±42
5??????+2=±42
&#3627409363;??????+&#3627409360;=&#3627409362;&#3627409360;
&#3627409363;??????=−&#3627409360;+&#3627409362;&#3627409360;
??????=
−&#3627409360;+&#3627409362;&#3627409360;
&#3627409363;
Divide both sides by 2
5??????+2
2
=32
&#3627409363;??????+&#3627409360;=−&#3627409362;&#3627409360;
&#3627409363;??????=−&#3627409360;−&#3627409362;&#3627409360;
??????=
−&#3627409360;−&#3627409362;&#3627409360;
&#3627409363;
&#3627409360;&#3627409363;??????+&#3627409360;
&#3627409360;
=&#3627409364;&#3627409362;
&#3627409360;