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Solving Equations Transformable to Quadratic Equation Including Rational Algebraic Equations
1. MATHEMATICS 9
Solving Equations Transformable to Quadratic
Equation Including Rational Algebraic
Equations
Lesson 1: Solving Quadratic Equations That Are Not
Written In Standard Form
Lesson 2: Solving Rational Algebraic Equations
Transformable To Quadratic Equations
2. In solving quadratic equation that is not written in standard form,
transform the equation in the standard form ax2 + bx + c = 0 where a,
b, and c are real numbers and a โ 0 and then, solve the equation
using any method in solving quadratic equation (extracting square
roots, factoring, completing the square, or quadratic formula).
Lesson 1: Solving Quadratic Equations That
Are Not Written In Standard Form
3. Example 1: Solve x(x โ 5) = 36.
Solution:
Transform the equation in standard form.
x(x โ 5) = 36
x2 โ 5x = 36
x2 โ 5x โ 36 = 0
Solve the equation using any method.
By factoring
x2 โ 5x โ 36 = 0 x โ 9 = 0 x + 4 = 0
(x โ 9)(x + 4) = 0 x = 9 x = โ 4
โข The solution set of the equation is {9, โ 4}.
4. Example 2: Solve (x + 5)2 + (x โ 2)2 = 37.
Solution:
Transform the equation in standard form.
(x + 5)2 + (x โ 2)2 = 37
x2 + 10x + 25 + x2 โ 4x + 4 = 37
2x2 + 6x + 29 = 37 โ 2x2 + 6x + 29 โ 37 = 0
2x2 + 6x โ 8 = 0
x2 + 3x โ 4 = 0 Divide all terms by 2.
Solve the equation using any method.
By factoring
x2 + 3x โ 4 = 0 x + 4 = 0 x โ 1 = 0
(x + 4)(x โ 1) = 0 x = โ 4 x = 1
โข The solution set of the equation is {โ 4, 1}.
5. Example 3: Solve 2x2 โ 5x = x2 + 14.
Solution:
Transform the equation in standard form.
2x2 โ 5x = x2 + 14
2x2 โ x2 โ 5x โ 14 = 0
x2 โ 5x โ 14 = 0
Solve the equation using any method.
By Quadratic Formula, identify the values of a, b, and c
a = 1, b = -5, c = -14
๐ฅ =
โ๐ ยฑ ๐2 โ 4๐๐
2๐
=
โ(โ5) ยฑ (โ5)2 โ4(1)(โ14)
2(1)
=
5 ยฑ 25 + 56
2
=
5 ยฑ 81
2
=
5 ยฑ 9
2
๐ฅ =
5+9
2
=
14
2
= ๐ ๐ฅ =
5โ9
2
=
โ4
2
= โ๐
The solution set of the equation is {โ 2, 7}.
6. Example 4: Solve (x โ 4)2 = 4.
Solution:
Transform the equation in standard form.
(x โ 4)2 = 4
x2 โ 8x + 16 = 4
x2 โ 8x + 16 โ 4 = 0
x2 โ 8x + 12 = 0
Solve the equation using any method.
By factoring
x2 โ 8x + 12 = 0 x โ 6 = 0 x โ 2 = 0
(x โ 6)(x โ 2) = 0 x = 6 x = 2
โข The solution set of the equation is {6, 2}.
7. Example 5: Solve (3x + 4)2 โ (x โ 1)2 = โ 5.
Solution:
Transform the equation in standard form.
(3x + 4)2 โ (x โ 1)2 = โ 5
9x2 + 24x + 16 โ (x2 โ 2x + 1) = โ 5
9x2 + 24x + 16 โ x2 + 2x โ 1 = โ 5
8x2 + 26x + 15 = โ 5 โ 8x2 + 26x + 15 + 5 = 0
8x2 + 26x + 20 = 0
4x2 + 13x + 10 = 0 Divide all terms by 2.
Solve the equation using any method.
By Quadratic Formula, identify the values of a, b, and c
a = 4, b = 13, c = 10
๐ฅ =
โ๐ ยฑ ๐2 โ 4๐๐
2๐
=
โ(13) ยฑ (13)2 โ4(4)(10)
2(4)
=
โ13 ยฑ 169 โ 160
8
=
โ13 ยฑ 9
8
=
โ13 ยฑ 3
8
๐ฅ =
โ13+3
8
=
โ10
8
= โ
๐
๐
๐ฅ =
โ13โ3
8
=
โ16
8
= โ๐
โข The solution set of the equation is {โ 2, โ
๐
๐
}.
8. Lesson 2: Solving Rational Algebraic Equations
Transformable To Quadratic Equations
There are rational equations that can be transformed
into quadratic equation of the form ax2 + bx + c = 0 where a,
b and c are real numbers, and a โ 0 and it can be solved
using the different methods in solving quadratic equation.
9. Steps in Solving Rational Equations:
1. Multiply both sides of the equation by the Least Common
Multiple (LCM) or Least Common Denominator (LCD).
2. Write the resulting quadratic equation in standard form.
3. Solve the equation using any method in solving quadratic
equation.
4. Check whether the obtained values of x satisfies the given
equation.
10. Example 1: Solve the rational algebraic equation
6
๐ฅ
+
๐ฅโ3
4
= 2.
Solution:
1. Multiply both side of the equation by the LCD, the LCD is 4x.
4๐ฅ
6
๐ฅ
+
๐ฅโ3
4
= 4๐ฅ(2) โ 4(6) + x(x โ 3) = 8x
24 + x2 โ 3x = 8x
2. Transform the resulting equation in standard form.
24 + x2 โ 3x = 8x โ x2 โ 3x โ 8x + 24 = 0
x2 โ 11x + 24 = 0
3. Solve the equation using any method. Since the equation is factorable,
x2 โ 11x + 24 = 0 x โ 3 = 0 x โ 8 = 0
(x โ 3)(x โ 8) = 0 x = 3 x = 8
The solution set of the equation is {3, 8}.
11. Example 2: Solve the rational algebraic equation
1
๐ฅ
+
1
๐ฅ+1
=
7
12
.
Solution:
1. Multiply both side of the equation by the LCD, the LCD is 12x(x + 1).
12๐ฅ(๐ฅ + 1)
1
๐ฅ
+
1
๐ฅ+1
= 12๐ฅ ๐ฅ + 1
7
12
โ 12(x + 1) + 12x = x(x+1)(7)
12x + 12 + 12x = 7x2 + 7x
2. Transform the resulting equation in standard form.
12x + 12 + 12x = 7x2 + 7x โ 0 = 7x2 + 7x โ 12x โ 12x โ 12
0 = 7x2 โ 17x โ 12
7x2 โ 17x โ 12 = 0
3. Solve the equation using any method. By Quadratic Formula, identify the values of a, b, and c. a = 7, b = โ 17, c = โ 12
๐ฅ =
โ๐ยฑ ๐2โ4๐๐
2๐
=
โ(โ17)ยฑ (โ17)2 โ4(7)(โ12)
2(7)
=
17ยฑ 289+336
14
=
17ยฑ 625
14
=
17ยฑ25
14
๐ฅ =
17+25
14
=
42
14
= ๐ ๐ฅ =
17โ25
14
=
โ8
14
= โ
๐
๐
โข The solution set of the equation is {3, โ
๐
๐
}.
12. Example 3: Solve the rational algebraic equation ๐ฅ +
8
๐ฅโ2
= 1 +
4๐ฅ
๐ฅโ2
.
Solution:
1. Multiply both side of the equation by the LCD, the LCD is x โ 2.
๐ฅ โ 2 ๐ฅ +
8
๐ฅโ2
= ๐ฅ โ 2 1 +
4๐ฅ
๐ฅโ2
โ x(x โ 2) + 8 = 1(x โ 2) + 4x
x2 โ 2x + 8 = x โ 2 + 4x
x2 โ 2x + 8 = 5x โ 2
2. Transform the resulting equation in standard form.
x2 โ 2x + 8 = 5x โ 2 โ x2 โ 2x โ 5x + 8 + 2 = 0
x2 โ 7x + 10 = 0
3. Solve the equation using any method. Since the equation is factorable,
x2 โ 7x + 10 = 0 x โ 5 = 0 x โ 2 = 0
(x โ 5)(x โ 2) = 0 x = 5 x = 2
โข The solution set of the equation is {5, 2}.
13. Example 4: Solve the rational algebraic equation
๐ฅ+3
3
+
1
๐ฅโ3
= 4.
Solution:
1. Multiply both side of the equation by the LCD, the LCD is 3(x โ 3).
3(๐ฅ โ 3)
๐ฅ+3
3
+
1
๐ฅโ3
= 3(๐ฅ โ 3)(4) โ (x โ 3)(x + 3) + 3(1) = 12(x โ 3)
x2 โ 9 + 3 = 12x โ 36
2. Transform the resulting equation in standard form.
x2 โ 6 = 12x โ 36 โ x2 โ 12x โ 6 + 36 = 0
x2 โ 12x + 30 = 0
3. Solve the equation using any method. By Quadratic Formula, identify the values of a, b, and c. a = 1, b = โ 12, c = 30
๐ฅ =
โ๐ยฑ ๐2โ4๐๐
2๐
=
โ(โ12)ยฑ (โ12)2 โ4(1)(30)
2(1)
=
12ยฑ 144โ120
2
=
12ยฑ 24
2
=
12ยฑ2 6
2
๐ฅ =
12+2 6
2
= ๐ + ๐ ๐ฅ =
12โ2 6
2
= ๐ โ ๐
โข The solution set of the equation is ๐ + ๐ , ๐ โ ๐ .
14. Example 4: Solve the rational algebraic equation
๐ฅ
3๐ฅ+2
=
2
๐ฅ+1
.
Solution:
1. Multiply both side of the equation by the LCD, the LCD is (3x + 2)(x + 1).
(3๐ฅ + 2)(๐ฅ + 1)
๐ฅ
3๐ฅ+2
= (3๐ฅ + 2)(๐ฅ + 1)
2
๐ฅ+1
โ x(x + 1) = 2(3x + 2)
x2 + x = 6x + 4
2. Transform the resulting equation in standard form.
x2 + x = 6x + 4 โ x2 + x โ 6x โ 4 = 0
x2 โ 5x โ 4 = 0
3. Solve the equation using any method. By Quadratic Formula, identify the values of a, b, and c. a = 1, b = โ 5, c = โ 4
๐ฅ =
โ๐ ยฑ ๐2 โ 4๐๐
2๐
=
โ(โ5) ยฑ (โ5)2 โ4(1)(โ4)
2(1)
=
5 ยฑ 25 + 16
2
=
5 ยฑ 41
2
๐ฅ =
๐+ ๐๐
๐
๐ฅ =
๐โ ๐๐
๐
โข The solution set of the equation is
๐+ ๐๐
๐
,
๐โ ๐๐
๐
.