Find the roots of the parabola given by the following equation.
2x2+ 5x - 9 = 2x

x=-3 or x=3/2

Step-by-step explanation:

We are given the following equation:

2x^2+5x-9=2x

We are asked to find the roots.  That means just solve it for x.

2x^2+5x-9=2x

Subtract 2x on both sides:

2x^2+3x-9=0

Let's see if we can put this in factored form.

Compare

2x^2+3x-9=0

and

ax^2+bx+c=0.

a=2, b=3 , c=-9

We have to find two numbers that multiply to be ac and add up to be b.

ac=-18

b=3

What are two numbers that multiply to be -18 and add to be 3?

Say -3 and 6.

So we are going to factor 2x^2-3x+6x-9=0

The first two terms have a common factor of x.

The last two terms have a common factor of 3.

2x^2-3x+6x-9=0

x(2x-3)+3(2x-3)=0

Now we can factor the (x-3) out of those 2 terms there since they share that common factor:

(x+3)(2x-3)=0

(x+3)(2x-3)=0 implies x+3=0  or 2x-3=0.

So we must solve x+3=0   and 2x-3=0

x+3=0

Subtract 3 on both sides:

x=-3

2x-3=0

Add 3 on both sides:

2x=3

Divide both sides by 2:

x=3/2

The solutions are x=3 or x=-3/2