Question

Which value for x makes the open sentence true? 7 + 6 • x = 42 + x2

Answer 1

The possible values of x are 3±√26i

Explanation:

Given the equation, 7 + 6x = 42 +x², to find the value of x that makes the expression true, we need to rearrange the expression and factorize the resulting equation.

Rearranging:

7 + 6x = 42 +x²

Moving 7+6x to the other side we have:

x²+42-6x-7 = 0

x²-6x+35 =0.

Using the general formula

x = -b±√b²-4ac/2a

From the quadratic function, a = 1, b= -6, c=35

x = 6±√(-6)²-4(1)(35)/2(1)

x = 6±√36-140/2

x = 6±√-104/2

x = (6±√104×-1)/2

x = 6±√104i/2

x = 6±2√26i/2

x = 3±√26i

Note that √-1 = I (a complex value)

Answer 2

x has 2 values:
`x_(1)=3 + √(26) *i` ,
`x_(2) = 3 - √(26)*i`

Explanation:

First: Rearrange the equation to get 0 as a result.

(In this case:
`x^(2) -6x + 42-7 = 0`

that can be reduced to:
`x^(2) -6x+35=0`)

Second: Apply the quadratic equation formula
`x = \frac{-b±\sqrt{b^(2)-4*a*c }}{2*a}`

For a= 1 ; b= -6 ; c= 35.

Third: Replace & solve to find
`x_(1)` and
`x_(2)` :

`(6±√(-104))/(2*1)` (Note:
`√(-104) = 2*√(-26)`)

And replacing
`√(-1) = i`:

`x_(1) = 3+√(26)*i` and
`x_(2) = 3-√(26)*i`

Note: If a character  appears, just ignore it (it is an error in the insertion of the formulas, it has no effect on the results)