SOLVING QUADRATIC TRINOMIAL BY TAKING SQUARE ROOTS

Solve the quadratic equation by taking square roots :

Problem 1 :

5x² - 20x + 20 = 35

Solution :

5x² - 20x + 20 = 35

x² - 4x + 4 = 7

x² - 4x + 4 - 7 = 0

x² - 4x - 3 = 0

x² - 2(x)(2) - 3 = 0

x² - 2(x)(2) + 2² - 2² - 3 = 0

Using the identity (a - b)² = a² - 2ab + b²

(x - 2)² - 2² - 3 = 0

(x - 2)² - 4 - 3 = 0

(x - 2)² - 7 = 0

(x - 2)² = 7

Take square root on both sides.

√ (x - 2)² = √7

x - 2 = ±√7

    x - 2 = √7        x - 2 = -√7

x = 2 + √7          x = 2 - √7

Therefore, the solutions are 

x = 2 + √7 or x = 2 - √7

Problem 2 :

x² - 2/3x + 1/9 = 1

Solution :

x² - 2/3x + 1/9 = 1

x² - 2/3x - 8/9 = 0

x² - 2(x)(1/3) - 8/9 = 0

x² - 2(x)(1/3) + (1/3)² - (1/3)² - 8/9 = 0

Using the identity (a - b)² = a² - 2ab + b²

(x - 1/3)² - (1/3)² - 8/9 = 0

(x - 1/3)² - (1/9) - 8/9 = 0

(x - 1/3)² - 1 = 0

(x - 1/3)² = 1

Take square root on both sides.

√(x - 1/3)² = √1

x - 1/3 = ±1

x - 1/3 = 1      x - 1/3 = -1

x = 4/3          x = -2/3

Therefore, the solutions are 

x = 4/3, x = -2/3

Problem 3 :

x² + 3/2x + 9/16 = 3

Solution :

x² + 3/2x + 9/16 = 3

x² + 3/2x + 9/16 - 3 = 0

 x² + 3/2x - 39/16 = 0

x² + 2(x)(3/4) - 39/16 = 0

x² + 2(x)(3/4) + (3/4)² - (3/4)² - 39/16 = 0

Using the identity (a + b)² = a² + 2ab + b²

(x + 3/4)² - (3/4)² - 39/16 = 0

(x + 3/4)² - (9/16) - 39/16 = 0

(x + 3/4)² - 3 = 0

(x + 3/4)² = 3

Take square root on both sides.

√(x + 3/4)² = √3

x + 3/4 = ±√3

x + 3/4 = √3      x + 3/4 = -√3

x = √3 - 3/4          x = -√3 - 3/4

Therefore, the solutions are 

x = ± √3 - 3/4

Problem 4 :

9x² + 12x + 4 = 5

Solution :

9x² + 12x + 4 = 5

9x² + 12x + 4 - 5 = 0

9x² + 12x - 1 = 0

x² + 4/3x - 1/9 = 0

x² + 2(x)(2/3) - 1/9 = 0

x² + 2(x)(2/3) + (2/3)² - (2/3)² - 1/9 = 0

Using the identity (a + b)² = a² + 2ab + b²

(x + 2/3)² - (2/3)² - 1/9 = 0

(x + 2/3)² - (4/9) - 1/9 = 0

(x + 2/3)² - 5/9 = 0

(x + 2/3)² = 5/9

Take square root on both sides.

√(x + 2/3)² = √5/9

x + 2/3 = ± √5/3

x = ± √5/3 - 2/3

Therefore, the solutions are 

x = √5/3 - 2/3 or x = -√5/3 - 2/3