## Related Calculator: Quadratic Equation Calculator

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.

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.

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.