4.2.2 - Quadratic Equations

4.2 - Algebra - Solving Equations

Leaving Certificate Mathematics

Ordinary Level & Higher Level

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 1 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x

(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x

− 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1)

(2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)

(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x

− 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

Answer:

12x2 − 10x + 10 = 8

12x2 − 10x + 10− 8 = 8− 8

12x2 − 10x + 2 = 0

6x2 − 5x + 1 = 0

6x2 − 3x − 2x + 1 = 0

3x(2x − 1) (2x − 1) = 0

(2x − 1)(3x − 1) = 0

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 2 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 1

Q. Solve for x: 12x2 − 10x + 10 = 8.

(2x − 1)(3x − 1) = 0

2x − 1 = 0

2x = 1

x = 12

3x − 1 = 0

3x = 1

= 13

∴ x = 13 ,

12

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 3 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer:

2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

2

2(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1

+3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1

=5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

2

2(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2)

+ 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2)

= 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1)

+ 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1)

= (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4

+ 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6

= 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1)

+ 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2

= 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x

+ 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

Q. Solve for x : 23x+1 + 3

x−1 = 52 .

Answer: 2

3x + 1+

3

x − 1=

5

22(3x + 1)(x − 1)(2)

3x + 1+

3(3x + 1)(x − 1)(2)

x − 1=

5(3x + 1)(x − 1)(2)

22(x − 1)(2) + 3(3x + 1)(2) = 5(3x + 1)(x − 1)

4(x − 1) + 6(3x + 1) = (15x + 5)(x − 1)

4x − 4 + 18x + 6 = 15x(x − 1) + 5(x − 1)

22x + 2 = 15x2 − 15x + 5x − 5

22x + 2 = 15x2 − 10x − 5

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 4 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

22x + 2 = 15x2 − 10x − 5

22x + 2− 15x2 + 10x + 5 = 0

− 15x2 + 32x + 7 = 0

x =−b ±

√b2 − 4ac

2a

=−32±

√322 − 4(−15)(7)2(−15)

=−32±

√1024 + 420

−30

=−32±

√1444

−30

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 5 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15

= −15

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6

Example 2

x =−32±

√1444

−30

=−32± 38

−30

=−16± 19

−15

x =−16 + 19

−15

=3

−15= −1

5

x =−16− 19

−15

=−35−15

= 73

∴ x = −15 ,

73

4.2 - Algebra - Solving Equations 4.2.2 - Quadratic Equations Ordinary Level & Higher Level 6 / 6