Incomplete Quadratic Equations

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Quadratic equation `ax^2+bx+c=0` is called incomplete, if either `b` or `c` (or both) equals 0.

Such equations can be easily solved without advanced methods.

Example 1. Solve `x^2-81=0`.

Here `a=1`, `b=0`, `c=-81`.

Add `81` to both sides to get equivalent equation `x^2=81`.

Now, let's carefully think.

What number, when squared, will give `81`?

One number is `9` (`9^2=81`), but there is another number: `-9` (`(-9)^2=81`).

Thus, to solve such equations, we need to take square root of both sides and don't forget, that there can be possibility for both plus and minus.

`x^2=81` becomes `sqrt(x^2)=+-sqrt(81)` or more simply `x=+-9`.

Answer: `x=9` and `x=-9`.

We can generalize result of this example.

Fact: roots of the equation `x^2=c` are `x=sqrt(c)` and `x=-sqrt(c)`.

However, if `c` is negative, then equation has no real roots (indeed, there is no such real number, that, when squared, will give negative number).

Example 2. Solve `2x^2+5=0`.

Subtract `5` from both sides: `2x^2=-5`.

Divide both sides by `2`: `x^2=-5/2`.

Since right hand side is negative, then equation has no roots.

Answer: no real roots.

Another case is when `c=0`.

Example 3. Solve incomplete quadratic equation: `3x^2-7x=0`.

Let's rewrite equation in the following form: `3x*x-7x=0`.

Now, rewrite equation, using distributive property of multiplication: `x(3x-7)=0`.

When is product of numbers equals 0?

When at least one factor equals 0.

So, either `x=0` or `3x-7=0`.

Second equation is linear, its root is `x=7/3`.

Answer: `x=0` and `x=7/3`.

We can generalize result of the above example.

Fact: incomplete quadratic equation `ax^2+bx=0` has two roots: `x=0` and `x=-b/a`.

Now, it is time to exercise.

Exercise 1. Find roots of the equation: `4y^2-25=0`.

Answer: `5/2` and `-5/2`.

Exercise 2. Solve the following: `3x^2=0`.

Answer: `0`. Hint: the only number, that, when squared, will give 0 is 0 itself.

Exercise 3. Solve `2z^2+7=0`.

Answer: no roots.

Exercise 4. Find roots of the quadratic equation `2x^2+7x=0`.

Answer: `0` and `-7/2`.

Exercise 5. Solve `-3z^2=-7`.

Answer: `sqrt(7/3)` and `-sqrt(7/3)`.

Exercise 6. Solve the incomplete quadratic equation `-4x^2-9=0`.

Answer: `0` and `-9/4`.