The Quadratic Formula
The method of completing the square can be applied lớn any quadratic polynomial.
You simply rewrite ax2+bx+c = a(x2+
From it we can obtain the following result:
The roots of ax2+bx+c are given by
The quantity b2−4ac is called the discriminant of the polynomial.
If b2−4ac the equation has no real number solutions, but it does have complex solutions. If b2−4ac = 0 the equation has a repeated real number root. If b2−4ac > 0 the equation has two distinct real number roots.Example
Study some of these examples:
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Find the roots of x2 + x + = 0
| x = ± sqrt( 2 − 4× × ) 2× x = | ± sqrt( | ) | |
| x = | , |
Example
x2 + x +
b2 - 4ac =
roots
x1 = , x2 =
Exercise
Now try some of these exercises:
The roots of x2 + x + are:
Working area:
Parabola Vertex
Note that if the roots of a quadratic equationax2+bx+c are real and distinct, then the vertex of the parabola given by the polynomial is situated where
Example
Study a few of these examples:
Locating the vertex of the parabola given by x2 + x + :
The x-coordinate is
| = |
2× |
Substituting this value of x into the given equation we find:
the y-coordinate is ( )2 + ( ) + =
Hence the vertex is ( , )
Exercise
Now try some of these exercises. Give your answers rounded khổng lồ 2 decimal places:
Locate the vertex of the parabola given by x2 + x + :
Working area:
The vertex is ( , )
If the roots of a quadratic equation ax2+bx+c are α và β, then we can write ax2+bx+c = a(x−α)(x−β)
Completing the Square | Quadratic Polynomials Index | Quadratic Functions Factoriser >>
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