Question

How do you solve `2x^{2}-28x+37=-111`?

Answer 1

No real solution. (i is an imaginary number)

Explanation:

`2x^2 -28x +37= -111`
Set all numbers to one side of the equation.
`2x^2 -28x +148= 0`
Divide by 2
`x^2 -14x +74= 0`
Use Quadratic Formula
`x=(\frac (-b\pm (\sqrt {b^{2)-4ac))){2a})`
Substitute
`x=(\frac (-(-14)\pm (\sqrt {(-14)^{2)-4(1)(74)))){2(1)})`
Solve
`x=(\frac (14\pm (\sqrt {(196)-296))){2(1)})`
`x=(\frac (14\pm (\sqrt {-100))){2(1)})`
`x=7\pm (10/2i)`
`x= 7+ 5i or x= 7- 5i`

Answer 2

`x=7+5i,` `7-5i`

Explanation:

Solve:

`2x^2-28x+37=-111`

Add `111` to both sides of the equation.

`2x^2-28x+37+111=0`

Simplify.

`2x^2-28x+148=0` is a quadratic equation in standard form:

`ax^2+bx+c=0`,

where:

`a=2`, `b=-28`, and `c=148`

Solve for `x` using the quadratic formula.

`x=(-b+-sqrt(b^2-4*a*c))/(2*a)`

Plug in the known values.

`x=(-(-28)+-sqrt(28^2-4*2*148))/(2*2)`

Simplify.

`x=(28+-sqrt(-400))/4`

Prime factorize `-400`.

`x=(28+-sqrt((2xx2)xx(2xx2)xx(5xx5)xx-1))/4`

Simplify.

`x=(28+-4xx5xxi)/4`

`x=(28+-20i)/4`

Factor out the common `4` in the numerator.

`x=(4(7+-5i))/4`

Divide `4` in the numerator by `4` in the denominator.

`x=7+-5i`

Solutions for `x`.

`x=7+5i,` `7-5i`