Equations
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Equations
(x + 2)^{2} = 3x + 6 is an example of an equation as the equality holds true for only certain values of x.
For example, if we substitute 1 for x in the left hand side and right hand side, we find:
The left hand side (LHS) = (1 + 2)^{2} = 3^{2} = 9
and the right hand side (RHS) = (3 x 1 + 6) = (3 + 6) = 9
But, if we substitute 3 for x in the left hand side and right hand side, we find the left hand side equals 25 but the right hand side only equals 15.
We must be able to solve linear, quadratic, and some more complex equations.
These are equations that can be arranged into the form ax + b = 0 and are solved by 'doing the same thing to both sides'.
For example:
3x − 2 = x + 12 (This solves to give x = 7)
These are equations the can be written in the form:
ax^{2} + bx + c = 0.
There are two types of quadratic equations that you may be asked to solve - those that factorise and those that do not.
Factorising
The principal reason for factorising is to help solve quadratics and other non-linear equations, using the idea that anything multiplied by zero is zero.
To do this we follow through the following stages.
1. Rewrite the equation in the form:
ax^{2} + bx + c = 0
2. Factorise.
3. Make each bracket equal to 0 to solve the equation.
Example:
x^{2} = 2x + 8 |
x^{2} − 2x − 8 = 0 |
(x − 4)(x + 2) = 0 |
therefore: x - 4 = 0, and x = 4,
or x + 2 = 0 and x = -2.
Completing the square
When we cannot factorise we can still solve a quadratic equation using a method called completing the square.
This works on the principal of rewriting the terms involving the variable x as a square number with an adjustment, and in doing so turns the equation into one that we can solve.
To do this we need to know that (x + a)^{2} = x^{2} + 2ax + a^{2}
- Rewrite in the form x^{2} + bx + c = 0, (if necessary divide by the multiple of x^{2}).
- Rewrite the x^{2} + bx as (x + b/2)^{2} − (b/2)^{2}. (Check to see if these are the same.)
- Now you can solve the equation.
Example:
Solve x^{2} = 2x + 7 (a similar equation to the earlier example)
First rearrange to get: x^{2} - 2x − 7 = 0
Rough Working for the solution:
so (finally!) x = 3.83 or x = -1.83
The Quadratic Formula
If we complete the square on the general equation, ax^{2} + bx + c = 0, we derive a quick formula for solving all quadratics. It is called the quadratic formula and is used when solving quadratics that do not factorise, (although it does work on ones that factorise too).
The formula is:
(See if you can derive this formula - it takes a lot of algebraic skill. If you can do it, it is a sure sign that your algebra skills are strong enough for anything A level can throw at you. Even if you can't don't worry! You can use the formula anyway!)
When you use the above equation you get 0, 1, or 2 roots (answers to the equation).
The number of solutions is determined by the value of b^{2} - 4ac, which has a special name 'the discriminant'.
A quadratic equation has:
Two real roots: | if b^{2} − 4ac > 0 |
Repeated roots: | if b^{2} − 4ac = 0 |
No real roots: | if b^{2} − 4ac < 0 |
Let's take an example equation, 2x^{2} + 7x + 3 = 0:
Comparing this to the standard format of a quadratic equation ax^{2} + bx + c = 0.
Gives a = 2, b = 7 and c = 3.
Putting this into the quadratic formula we get:
Note: As the answers are rational we could have solved this by factorising:
(2x +1)(x + 3) = 0
x = -0.5 or x = -3
The solutions of a quadratic equation and the original equation are linked. If you know the solution of the quadratic equation, the original equation can be found using the equation below:
x^{2} − (sum of roots) x + product of roots = 0
Lets call the solution of the equation ax^{2} + bx + c = 0, α and β. Then the
Sum of roots (α + β) = -b/a
product of roots (α β) = c/a
Therefore, if you know that the sum of the roots is 6 and the product is 12, you can write the quadratic equation as:
x^{2} − 6x + 12 = 0.