A quadratic equation in which `a` (the leading coefficient) is negative can be solved in two ways:

### Multiply the equation throughout by -1

On multiplying a negative number with -1, you get a positive number. So on multiplying a quadratic equation with -1, the sign on `a` will get changed. For example `-1 * (-3x^2 + 2x + 1 = 0)` gives you `3x^2 - 2x - 1 = 0`.

Now you can solve the equation by any of the methods of factoring, completing the square, or by applying the quadratic formula.

Note that you will get the same answer for a quadratic equation on multiplying it throughout by -1 as with solving it without doing so.

### Solve the equation as it is

A quadratic equation can be solved regardless of whether `a` is negative or not. You can solve it by factoring, completing the square, or applying the quadratic formula.

For factoring a quadratic expression when `a` is negative, all you need to do is remember to `a` as negative when multiplying it with `c` to get their product. Then find two numbers whose product is equal to the above product and whose sum is `b`, the coefficient of the middle term or `x` in the equation.

In this process you should take the signs on the left of each term `a`, `b`, or `c`. For example, `-3x^2 + 2x + 1` can be factored as follows:

- a = -3
- b = 2
- c = 1

Product of `a` and `c = -3 * 1 = -3`. Two numbers whose product is -3 and sum is 2 are 3 and -1. So factor the quadratic expression as follows:

`-3x^2 + 3x - x + 1`

`-3x(x - 1) - 1(x - 1)`

`(x - 1)(-3x - 1)`

Some quadratic equations may not factor, so these can be solved by the other two methods mentioned above.

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