Question

How do you solve using the quadratic formula `6x+9=2x^2`?

Answer 1

`x=3/2+3/2 sqrt 3` or `x=3/2-3/2 sqrt 3`

Explanation:

`6x+9=2x^2`

`:.2x^2=6x+9`

`:.2x^2-6x-9=0`

Quadratic formula:- `x=ax^2+bx+c`

`:.a=2,b=-6 and c=-9`

`:.x=(-b+-sqrt (b^2-4ac))/(2a)`

`:.x=(-(-6)+-sqrt((-6^2)-4(2)(-9)))/(2(2))`

`:.x=(6+-sqrt(108))/4`

`:.x=(6+-sqrt(3*3*3*2*2))/4`

`:.sqrt2*sqrt2=2`

`:.sqrt 3*sqrt 3=3`

`:.x=(6+-3*2 sqrt 3)/4`

`:.x=(6+-6 sqrt 3)/4`

`:.x=cancel 6^3/cancel 4^2+-(cancel 6^3 sqrt 3)/cancel 4^2`

`:.x=3/2+-(3 sqrt 3)/2`

`:.x=3/2+3/2 sqrt 3` or `x=3/2-3/2 sqrt 3`

Answer 2

`x=3/2+−3/2sqrt(3) or x=3/2+3/2sqrt3`

Explanation:

Step 1: Subtract `2x^2` from both sides.
`6x+9−color(blue)(2x^2)=2x^2−color(blue)(2x^2)`
`−2x^2+6x+9=0`
Step 2: Use quadratic formula with `color(red)(a=-2), color(purple)(b=6), color(green)(c=9).`
`x=−b±sqrt(b^2−4ac)/(2a)`
`x=−(color(purple)(6))^2±sqrt((color(purple)(6))^2−4(color(red)(−2))(color(green)(9)))/(2(color(red)(−2))`
x=
`−6±sqrt108/(−4)`
`x=3/2+−3/2sqrt3` or #x=3/2+3/2 sqrt3#