Question

How do you solve `2x^2 - x = 6` graphically and algebraically?

Answer 1

`x=2, and -3/2`
`-3/2 "can also be written as" -1.5`

Explanation:

Let's draw graph of the expression using graphing tool.
graph{2x^2-x-6 [-9.86, 10.14, -7.64, 2.36]}
Observe in the graph where it intersects `x` axis.
`(-1.5,0) and (2,0)`
`-1.5 and 2` are solutions of the equation.

Given equation is `2x^2-x=6`.
Taking all terms to left hand side we get
`2x^2-x-6=0`
Let's try split the middle term method.
Product of first and third term `=-12x^2`
Let's find two parts of middle term so that their product is equal to above. `12` has factors of `6 xx 2, 4 xx 3, 12 xx 1`. We see that `4xx3` parts work for us. Splitting middle term as in two parts `-4x and 3x` we obtain

`2x^2-4x+3x-6=0`, pair first two and last two
`(2x^2-4x)+(3x-6)=0`, Take out Common factor `2x` out of first pair and `3` out of second pair.
`2x(x-2)+3(x-2)=0`, Take out Common factor `x-2` out of two
`(x-2)(2x+3)=0`, Set each factor equal to `0`
`(x-2)=0` .......(1)
`(2x+3)=0` .....(2)
From (1) `x=2`
From (2) `x=-3/2`