In this unit, we'll see how we can use the imaginary number I
to solve any quadratic equation.
Let's have a look at an example. Suppose we want to solve the
quadratic equation X squared
minus 2X. Plus 10
equals 0. Now we're going to use the formula for solving a
quadratic equation. This is the formula over here and first of
all we need to identify the values of AB&C to substitute in
the formula. Now the value the value of a is the coefficient of
X squared, which in this case is
one. The value of B is the coefficient of X, which is minus
2. And the value of C is the constant term, which is 10.
Can we substitute these values into this formula? So here we go
X equals. Minus B, which is minus minus two, which is +2.
Plus or minus the square root of B squared, B squared is minus 2
squared, which is +4.
Minus four times a, which was one and see which was 10.
All divided by 2A and 2A is 2 one or two.
Let's tidy this up. We've got 2 plus or minus. Now let's look
under the square root sign. We've got 4. Subtract 4 * 1 *
10. 4 * 1 * 10 is 44. Subtract 40 is minus 36, so you'll see
we've got a square root of a negative number. Here. The
square root of minus 36, and it's all divided by two.
Let me remind you how you deal with the square root of a
negative number. The square root of minus 36. We can write as the
square root of 36 times minus
one. Which is 6.
Times I or six I.
So the square root of this negative number, the square root
of minus 36, simplifies to just
six I. And finally, if we just want to tidy this up a bit more,
we can notice that there's a factor of two in the numerator
and the denominator, which can be cancelled, which will leave
one plus or minus three. I, so here we have two solutions of
the quadratic equation. One of the Solutions is is the number 1
+ 3 I and another is the number 1 - 3 I.
Let's have a look at another example. In this example, we're
going to study the quadratic equation, two X squared plus X
Plus One is 0.
Again, in order to
use the
formula. Which is here we need to identify the values of AB&C.
The value of a, which is the coefficient of X squared, is 2.
The value of B is one and the value of C is also one.
And we substitute these values into the formula.
So we'll get X equals minus B, which is minus one plus or minus
the square root of be squared.
Which is 1 squared.
Minus four times a which was two and see which was one all
divided by. To a which is 2 twos of four.
Let's tidy it. What we've got minus one plus or minus. Now
let's look at the square root. We've got 1 squared, which is
one subtract 428, so it's one subtract 8, which is minus
Seven. You'll see again that we've ended up with a square
root of a negative number.
Now the square root of minus Seven we handle in the same way
as before. We write it as the square root of 7 times minus
one. The square root of Seven relievers. The square root of 7
and the square root of minus one. We now right as I.
So this solution we have here now simplifies to minus one plus
or minus the square root of minus Seven. We write as the
square root of 7.
Times, I and the whole things divided by 4 and we can leave
our answer like that. But if we want to we can write it as
separate terms. We can write it is minus 1 / 4.
Plus or minus.
The square root of 7.
Divided by 4 multiplied by I. So either of those
forms are equivalent.
We've now seen how we can write down the solution of
any quadratic equation.
A number such as this one which has got a part which is purely
real in this case, minus 1/4 and a part which is imaginary.
That's the part that this number here. Route 7 over 4 multiplied
by this imaginary number I a number such as this is called a
complex number, and in the next unit will define properly what
we mean by a complex number.