Question

What are the approximate solutions of `2x^2 + x = 14` rounded to the nearest hundredth?

Answer 1

`color(green)(x=2.41` or `color(green)(x=-2.91)color(white)("xxx")`(both to the nearest hundrdeth.

Explanation:

Re-writing the given equation as
`color(white)("XXX")color(red)2x^2+color(blue)1xcolor(green)(-14)=0`
and applying the quadratic formula:
`color(white)("XXX")x=(-color(blue)1+-sqrt(color(blue)1^2-4 * color(red)2 * color(green)(""(-14))))/(2 * color(red)2)`

`color(white)("XXXx")=(-1+-sqrt(113))/4`

with the use of a calculator (or, im my case I used a spreadsheet)
`color(white)("XXX")x ~~ 2.407536453color(white)("xxx")orcolor(white)("xxx')x~~-2.9075366453`

Rounding to the nearest hundredths gives the results in the "Answer" (above)

Answer 2

See a solution process below:

Explanation:

First, we can subtract `color(red)(14)` from each side of the equation to put the equation in standard form while keeping the equation balanced:

`2x^2 + x - color(red)(14) = 14 - color(red)(14)`

`2x^2 + x - 14 = 0`

We can now use the quadratic equation to solve this problem.

The quadratic formula states:

For `color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0`, the values of `x` which are the solutions to the equation are given by:

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

Substituting:

`color(red)(2)` for `color(red)(a)`

`color(blue)(1)` for `color(blue)(b)`

`color(green)(-14)` for `color(green)(c)` gives:

`x = (-color(blue)(1) +- sqrt(color(blue)(1)^2 - (4 * color(red)(2) * color(green)(-14))))/(2 * color(red)(2))`

`x = (-color(blue)(1) +- sqrt(1 - (-112)))/4`

`x = (-color(blue)(1) +- sqrt(1 + 112))/4`

`x = (-color(blue)(1) - sqrt(1 + 112))/4` and `x = (-color(blue)(1) + sqrt(1 + 112))/4`

`x = (-color(blue)(1) - sqrt(113))/4` and `x = (-color(blue)(1) + sqrt(113))/4`

`x = (-color(blue)(1) - 10.6301)/4` and `x = (-color(blue)(1) + 10.6301)/4`

`x = -11.6301/4` and `x = 9.6301/4`

`x = -2.91` and `x = 2.41` rounded to the nearest hundredth.