Solve by completing the square method

Question

1. x^2-10x+15=0
2. 3x^2+4x=6

Answer 1

To solve the equations using the completing the square method, follow these steps:

1. x^2 - 10x + 15 = 0

To complete the square, we need to add a constant term to both sides of the equation such that the left side can be factored as a perfect square.

First, write the equation in the form (x - a)^2 + b = 0.

x^2 - 10x + 15 = (x - 5)^2 - 25 + 15 = (x - 5)^2 - 10 = 0

Since the equation is equal to 0, we subtract 10 from both sides:

(x - 5)^2 = 10

Taking the square root of both sides:

x - 5 = ±√10

Adding 5 to both sides:

x = 5 ± √10

Therefore, the solutions are x = 5 + √10 and x = 5 - √10.

2. 3x^2 + 4x = 6

To complete the square, we divide the entire equation by the coefficient of x^2, which is 3.

x^2 + (4/3)x = 2

To find the constant to complete the square, take half the coefficient of x and square it - in this case, it would be (4/6)^2 = 4/9.

Add the constant to both sides of the equation:

x^2 + (4/3)x + 4/9 = 2 + 4/9

To factor the left side as a perfect square, write it as (x + a)^2, where a is half the coefficient of x.

(x + 2/3)^2 = 2 + 36/9 - 4/9
(x + 2/3)^2 = 2 + 32/9
(x + 2/3)^2 = 50/9

Taking the square root of both sides:

x + 2/3 = ±√(50/9)

x + 2/3 = ±(5√2)/3

Subtracting 2/3 from both sides:

x = -2/3 ± (5√2)/3

Therefore, the solutions are x = (-2 + 5√2)/3 and x = (-2 - 5√2)/3.