The Quadratic Formula
Question 20 - Solve the equation using the
quadratic formula: 2x squared equals 8X plus
3.
In order to use the quadratic formula, we
need to write the equation in Ax squared plus
BX plus C form.
To do that, I'm going to subtract the 8x and
the 3 from both sides.
This will leave me with 2x squared minus 8x
minus 3 equals 0.
Now I need to identify the values for A, B,
and C to use in my quadratic formula, which
is listed here.
The value for A is the number in front of
the x squared term.
So, A equals 2.
The value for B is the number in front of
the X term.
Remember, we need to take the sign with it,
so B equals negative 8.
The value for C is the constant term and we
need to also take the sign with it, so C equals
negative 3.
Next, we're going to plug the values in for
A, B, and C into the quadratic formula and
solve.
I have a negative B which is a negative, negative
8 plus or minus the square root of B squared.
So, negative 8 squared minus 4 times A, which
is 2, times C, which is negative 3, and that
is all over 2A, which is 2 times 2.
Now we need to simplify this.
A negative of a negative 8 turns into a positive
8.
Then plus or minus the square root of negative
8 squared.
Negative 8 squared is negative 8 times negative
8, which gives us 64.
Now we're going to subtract 4 times 2 times
negative 3.
4 times 2 is 8 and 8 times negative 3 is negative
24.
So, I have 64 minus a negative 24 and this
is all being divided by 2 times 2, which is
4.
Now I'm going to simplify what's underneath
the square root bar.
So, I have 8 plus or minus the square root
of 64 minus a negative 24.
Whenever you have two minus signs together
or you're subtracting a negative that turns
into a positive.
So, this is 64 plus 24 and 64 plus 24 is 88.
Next, I want to try and reduce the square
root of 88.
In order to do that, I'm going to list the
prime factors of 88.
So, 8 times 11 is 88 and 4 times 2 gives me
8 and then 2 times 2 gives me 4.
I'm just writing a factor tree of all the
prime factors and I factored those numbers
till they couldn't be factored anymore, so
my resulting factors are 2 times 2 times 2
times 11.
I can replace the 88 with 2 times 2 times
2 times 11.
Now the square root operation tells me to
look for pairs of two.
I have a pair of twos underneath the bar.
When I pull the pair out, it becomes a single
digit.
So, just the number 2 comes out.
Now I have 8 plus or minus 2 square root of
2 times 11, because that's what's left after
I took the square root of that pair.
This is all being divided by 4.
We can simplify that further by multiplying
the values underneath the square root bar.
So, I have 8 plus or minus 2 times the square
root of 22 over 4.
I'm almost done, but the last thing I can
do is try to simplify the coefficients by
the denominator.
My coefficients are the 8 and the 2.
So 8, 2, and 4 are all divisible by 2.
That tells me that I can reduce all of those
numbers by 2.
So, I'll divide each of those numbers by 2
to get my final result.
Now 8 divided by 2 is 4.
So, this becomes a 4 plus or minus 2 divided
by 2 is 1, so I can just leave the square
root of 22 there and then 4 divided by 2 is
2, so a 2 goes on the bottom.
This is my final answer: x equals 4 plus or
minus the square root of 22 all over 2 and
again we can write those as two separate solutions
by separating the plus and minus sign.
This would give me x equals 4 plus the square
root of 22 over 2 and x equals 4 minus the
square root of 22 over 2.
