Question 245016: Iam preparing for GcSE
and i still dont get it can you give me some help or useful information which could help me
Found 2 solutions by Theo, solver91311:
Answer by Theo(13342)   (Show Source):
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
I'm going to go out on a limb here and assume that what you don't get is the concept of quadratic equations and the process of solving them.
A quadratic equation is an equation in the form
Typically,  ,  , and  are integers, or at the very least rational, in the problems that preparatory school algebra students are asked to solve. However, they don't even have to be real numbers. The only restriction is that must be non-zero -- otherwise you do not have a quadratic.
In accordance with the Fundamental Theorem of Algebra which tells us that there are always  roots of a polynomial equation of degree , we know that every equation of the form  must have two solutions, aka roots.
There are three methods for solving a quadratic equation.
1. Factoring. If you can determine numbers  ,  ,  , and  such that:
and
then you can write:
(rx\ +\ s)\ =\ 0)
Knowing that you can apply the Zero Product Rule which says:
to write:
or
In the event that , , , and/or are irrational or complex numbers, finding these numbers could prove an insurmountable task for the average solver. This leads us to the two other methods.
2. Completing The Square. The discussion of this method in general terms will lead us directly to the third method, namely, the Quadratic Formula.
Given
Proceed step-by-step as follows:
Add the additive inverse of the constant term to both sides:
Multiply both sides by the multiplicative inverse of the lead coefficient:
Divide the first degree term coefficient by 2, square the result, and add the result to both sides of the equation:
Note that the LHS is now a perfect square trinomial (hence the name of the method: Completing the Square)
Factor the LHS:
^2\ =\ \frac{b^2}{4a^2}\ -\ \frac{c}{a})
Apply the LCD to the RHS and combine:
^2\ =\ \frac{b^2\ -\ 4ac}{4a^2})
Take the square root of both sides:
Then finish solving for 
You can perform the above procedure using the specific coefficients in the equation you are trying to solve, or you can simply substitute your coefficients into the resulting formula which is known as the quadratic formula.
3. Quadratic Formula
The quadratic formula is simply the final result of Completing the Square shown above. Simply substitute the values of the coefficients from your standard form quadratic equation, do the arithmetic, et voilą.
The part of the quadratic formula that is under the radical, namely  , is known as the discriminant. The value of the discriminant relative to zero tells you about the character of the roots of your equation. While the fundamental theorem of algebra does guarantee that a quadratic will have two roots, it does not guarantee that the roots will be real numbers.
Let:
Then if:
Then the roots of the equation are two distinct real numbers.
But if:
Then the roots of the equation are two equal real numbers. Some texts say there is one root with a multiplicity of two, and also couch the fundamental theorem in terms of counting the roots up to the limit of any multiplicities.
And finally, if:
Then the roots of the equation are a conjugate pair of complex numbers of the form  where  is the imaginary number defined by  . Since we know that complex number roots of quadratic equations always come in conjugate pairs, if we are given that  is a root, then  must also be a root.
Quadratic equations have a strong relationship with quadratic functions.
Equation:
Function:
\ =\ ax^2\ +\ bx\ +\ c)
The graph of a quadratic function is a parabola. It opens upward if is positive and downward if is negative. It has a vertex at:
\right))
The vertex is a minimum value for the function if the parabola opens upward and a maximum value if it opens downward.
The graph intersects the  -axis at \right))
The graph intersects the -axis at ) and ) where  and  are the roots of the corresponding quadratic equation -- IF the roots are real. If the roots are complex, then the graph of the quadratic does not intersect the -axis at all. In the case where the two roots are real and equal, the -axis intersection is the vertex of the parabola.
A common example of a quadratic function is a projectile launched vertically from the surface of the earth, with initial velocity  and initial height of  . The acceleration due to gravity near the surface of the earth being  or  , depending on your desired units of measurement.
The height function is:
\ = -\frac{1}{2}gt^2\ +\ v_ot\ +\ h_o)
Notice that this would be a parabola opening downward. Therefore the value of the function at the vertex would be the maximum value of the function. Hence, the elapsed time, in seconds, until the projectile reaches its maximum height would be given by
And the actual maximum height achieved would be
\ =\ -\frac{1}{2}g\left(\frac{v_o}{g}\right)^2\ +\ v_o\left(\frac{v_o}{g}\right)\ +\ h_o\ =\ \frac{v_o^2}{2g}\ +\ h_o)
The projectile will return to earth at time equal to the positive root of:
Note that if  , then one of the roots will be time 0. If  , then there will be two roots, one of which will be negative.
What else do you want to know?
John
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