Question

X² : (x + 6) = 1 : 2
Find the possible values of x

Answer 1

x= 2 and x= -(2/3)

Explanation:

Step 1 :Multiply all terms by the same value to eliminate fraction denominators

x^2/ x+6 = 1/2

2(x+6)*(x^2/x+6) = 2(x+6) * 1/2

Step 2: Cancel multiplied terms that are in the denominator

2x^2= x+6

Step 3: Move terms to the left side

2x^2 - (x+ 6) = 0

Step 4: Distribute

2x^2 - x - 6 = 0

Step 5: Use the quadratic formula

x= ( - (-1) +- Squar((-1)^2-4*2(-6)) ) / 2*2

Step 6: Simplify

x= (1+- 7)/ 4

Step7: Separate the equations

x= (1+7)/ 4 and x= (1- 7)/ 4

Step 8: Solve that 2 small equation

x= 2 and x= -(3/2)

Answer 2

x = -
`(3)/(2)` , x = 2

Explanation:

x² : x + 6 = 1 : 2

express the ratio in fractional form

`(x^2)/(x+6)` =
`(1)/(2)` ( cross- multiply )

2x² = x + 6 ( subtract x + 6 from both sides )

2x² - x - 6 = 0 ← in standard form

consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.

product = 2 × - 6 = - 12 and sum = - 1

the factors are - 4 and + 3

use these factors to split the x- term

2x² - 4x + 3x - 6 = 0 ( factor the first/second and third/fourth terms )

2x(x - 2) + 3(x - 2) = 0 ← factor out (x - 2) from each term

(x - 2)(2x + 3) = 0 ← in factored form

equate each factor to zero and solve for x

x - 2 = 0 ⇒ x = 2

2x + 3 = 0 ⇒ 2x = - 3 ⇒ x = -
`(3)/(2)`