How does square root property and real numbers relate to quadratic equations?
To solve quadratic equations, it is necessary to use square roots. For example, to solve the equation
`x^2 = 4`
one has to take square root of both sides:
`sqrt(x^2) = sqrt(4)`
The property of square root is that `sqrt(x^2) = |x|`
|x| = 2
`x = +-2`
(x can be positive or negative 2 to satisfy the equation `x^2 = 4` ) This quadratic equation has 2 real solutions.
For more complicated quadratic equations, such as `x^2 + 6x +8=0` , the solution procedure is longer but the principle remains the same. This quadratic equation also has two real solutions.
But the quadratic equation such as
does not have any real solutions because it is impossible to take a square root of a negative number:
`sqrt(-4) ` is not a real number.
For general quadratic equation in the form `ax^2 + bx + c = 0`
the solutions are given by quadratic formula:
`x_(1,2) = (-b+-sqrt(b^2 - 4ac))/(2a)`
The expression under the radical, `b^2 - 4ac` , is called "discriminant". If it is positive, the quadratic equation has two real solutions. If it 0, the equation has 1 real solution. If it is negative, the equation has no real solutions.