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### Quadratic equations

A quadratic equation involves and when we look to solve an equation of this sort we need to find the value of the unknown *x*. Because the unknown could really be any value, trial and error is not a very good method to use as it will take a huge amount of time. Therefore, we need a method to use in order to solve an equation of the type

By realising that two unknowns when multiplied together equals 0, one of the unknowns must be 0 itself. This is because the only way for *a* or *b* to be 0 (or both of *a* and *b* to be 0). Therefore, by factorising the equation into two different brackets multiplied together, we can see that one or both of these brackets must equal 0.

#### Example

Solve the following equations:

1)

2)

1) This equation has two distinct parts which are multiplied to get 0. This means that one of the two brackets MUST be equal to zero. So we can now just work out the value of *x* in either case.

2) For this equation we must first factorise into double brackets using the method that we have already seen in this lesson. To get a product of 7 we must have the pair of numbers 7 and 1, of which one must be negative since we really have

Now we can solve by setting either bracket to equal 0.

So

Whenever we get an equation that contains

This means that if we have something like

The same can be done for something like

#### Example

Solve the following quadratic equations for the unknown value of *x*.

1)

2)

3)

1) Here we are missing a value for in the equation so the must be 0. This leaves us with

This then gives us two values for when either bracket is equal to zero. So we have an answer of

2) Putting this into the equation we have

3) With this equation we can immediately divide the entire thing by 2 so that we have

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