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1. Grade 9 – Mathematics
Quarter I
SOLVING QUADRATIC EQUATIONS BY
COMPLETING THE SQUARE

2. OBJECTIVES:
•Express square of trinomials
as a square of binomials; and
•Solve quadratic equations by
completing the square.

3. COMPLETING THE SQUARE
To complete the square of the
expression 𝑥2
+ 𝑏𝑥, add the square of
half the coefficient of 𝑥 to make
𝒙 𝟐
+𝒃𝒙 +
𝒃
𝟐
𝟐
.

4. Complete the square.
𝒙 𝟐
− 𝟖𝒙 + ______
𝒙 𝟐
+ +𝒃𝒙 +
𝒃
𝟐
𝟐
−
8
2
= −4
(−4)2
= 16
𝒙 𝟐
− 𝟖𝒙 + 𝟏𝟔 (𝒙 − 𝟒) 𝟐

5. Complete the square.
𝒙 𝟐
+ 𝟑𝒙 + ______
𝒙 𝟐
+ +𝒃𝒙 +
𝒃
𝟐
𝟐
3
2
=
3
2
3
2
2
=
9
4
𝒙 𝟐
+ 𝟑𝒙 +
𝟗
𝟒
𝑥 +
3
2
2

6. Express each of the following perfect square trinomials
as a square of a binomial.
1. 𝑥2
+ 4𝑥 + 4 (𝑥 + 2)2
2. 𝑥2
+ 12𝑥 + 36 (𝑥 + 6)2
3. 𝑥2
− 10𝑥 + 25 (𝑥 − 5)2
4. 𝑥2
− 5𝑥 +
25
4
(𝑥 −
5
2
)2

7. 𝑥2
+ 10𝑥 = 3
𝑥2
+ 10𝑥 + ______
10
2
= 5
(5)2
= 25
𝑥2
+ 10𝑥 +25
(𝒙 + 𝟓) 𝟐
𝑥2
+ 10𝑥 + 25 = 3 + 25
Add 25 to both sides of the equation
Express the PST on the left side of the equation as square of binomial.
(𝑥 + 5)2
= 28
Solve the resulting QE by extracting the square root.
𝑥 + 5 = ± 28
Solve each equation.
𝑥 + 5 = 2 7 𝑥 + 5 = −2 7
𝑥 = −5 + 2 7 𝑥 = −5 − 2 7
𝑥 + 5 = ±2 7
28 = 4 ∙ 7 = 2 7

8. 𝑥2 − 3𝑥 = 1𝑥2
− 3𝑥 + ______
−
3
2
−
3
2
2
=
9
4
𝑥2
− 3𝑥 +
9
4
𝑥2
− 3𝑥 +
9
4
= 1 +
9
4
Add
9
4
to both sides
of the equation
Express the PST on the left
side of the equation as square
of binomial.
x −
3
2
2
=
13
4
Solve the resulting QE by
extracting the square root.
𝒙 −
𝟑
𝟐
𝟐
𝑥 −
3
2
= ±
13
4
𝑥 −
3
2
= ±
13
2

9. Solve each equation.
𝑥 −
3
2
=
13
2
𝑥 −
3
2
= −
13
2
𝑥 =
3
2
+
13
2
𝑥 =
3
2
−
13
2
𝒙 =
𝟑 + 𝟏𝟑
𝟐
𝒙 =
𝟑 − 𝟏𝟑
𝟐
𝑥 −
3
2
= ±
13
2

10. 𝑥2
+ 5𝑥 + 4 = 0
𝑥2
+ 5𝑥 + _____
5
2
5
2
2
=
25
4
𝑥2
+ 5𝑥 +
25
4
𝑥2
+ 5𝑥 +
25
4
= −4 +
25
4
Add
25
4
to both sides of the
equation
Express the PST on the left side of the
equation as square of binomial.
x +
5
2
2
=
9
4
Solve the resulting QE by
extracting the square
root.
𝒙 +
𝟓
𝟐
𝟐
𝑥 +
5
2
= ±
9
4
𝑥 +
5
2
= ±
3
2
𝑥2
+ 5𝑥 = −4

11. Solve each equation.
𝑥 +
5
2
=
3
2
𝑥 +
5
2
= −
3
2
𝑥 = −
5
2
+
3
2
𝒙 = −𝟏
𝑥 +
5
2
= ±
3
2
𝑥 = −
2
2
𝑥 = −
5
2
−
3
2
𝒙 = −𝟒
𝑥 = −
8
2

12. 2𝑥2
− 3𝑥 − 9 = 0
Add
9
16
to both sides of the equation
Express the PST on the left side
of the equation as square of binomial.
Solve the resulting QE by
extracting the square root.
2𝑥2
− 3𝑥 = 9
Divide the terms by 2. 𝑥2
−
3𝑥
2
=
9
2
−
3
2
2
= −
3
2
∙
1
2
= −
3
4
−
3
4
2
=
9
16
𝑥2 −
3𝑥
2
+
9
16
𝒙 −
𝟑
𝟒
𝟐
𝑥2
−
3𝑥
2
+
9
16
=
9
2
+
9
16
𝑥 −
3
4
2
=
81
16
𝑥 −
3
4
= ±
81
16
𝑥 −
3
4
= ±
9
4

13. Solve each equation.
𝑥 =
3
4
+
9
4
𝑥 −
3
4
=
9
4
𝑥 −
3
4
= ±
9
4
𝑥 −
3
4
= −
9
4
𝑥 =
3
4
−
9
4
𝑥 =
−6
4𝑥 =
12
4
𝒙 = 𝟑 𝒙 = −
𝟑
𝟐