To solve the equation 2x² - 4 = 3x, we need
to first identify what type of equation this
is. Looking through the equation, I don’t
see any radicals, so this is not a radical
equation. I do see that there is a variable
that is squared, this x² term. Knowing that
this is an x², and that is the highest degree
in the equation tells me that this is a quadratic
equation. The methods we know right now to
solve quadratic equations are the square root
property and the quadratic formula. Because
we have this 3x term over on the right side
of the equation, I recognize that I cannot
use the square root property. The square root
property only works if there is a single variable
caught up in what is being squared, and there’s
only numbers otherwise. So in this case, we’re
going to need to use the quadratic formula.
As a reminder, the quadratic formula is based
on this: If ax² + bx + c = 0, then x is equal
to the opposite of b plus or minus the square
root of b² - 4ac all over 2a. The quadratic
formula allows us to solve any quadratic equation,
but that equation must be set equal to 0.
So our first step in solving 2x² - 4 = 3x
will be to set that equation equal to 0 by
subtracting 3x from both sides and putting
it in standard form. Subtracting 3x from both
sides and putting it in the standard form
would then give us the equation 2x² - 3x
- 4 = 0. And now that it is in that standard
ax² + bx + c = 0 form, I can identify my
a, my b, and my c for the quadratic formula.
My a is 2, my b is -3, and my c is -4. So
when I go to set up the quadratic formula,
I will have x equals the opposite of -3 plus
or minus the square root of (-3)² - 4(2)(-4),
and all of that over 2(2). The first steps
in simplifying this will involve cleaning
up the opposite of -3, becoming 3 in the front.
Underneath the radical, -3² is 9. Also under
the radical, -4(2)(-4) is +32. And 2(2) in
the denominator is 4, giving us 3 plus or
minus the square root of 9 + 32 all over 4.
Continuing to simplify that expression, we
can clean up more under the radical. 9 + 32
is 41. And so we have 3 plus or minus the
square root of 41 all over 4. Anytime we’re
working with radicals, we need to ask ourselves
if we can reduce or simplify that radical.
In this case, 41 is underneath the square
root, and there is no perfect square that
divides evenly into 41, and so we have our
two solutions, 3 plus root 41 all over 4 and
3 minus root 41 all over 4. So we can write
that as a conclusion, the solution set is
the set 3 plus or minus root 41 all over 4.
