Incomplete Quadratic Equations

Quadratic equation $$${a}{{x}}^{{2}}+{b}{x}+{c}={0}$$$ 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 $$$9$$$ $$$\left({{9}}^{{2}}={81}\right)$$$, but there is another number: $$$-{9}$$$ $$$\left({{\left(-{9}\right)}}^{{2}}=81\right)$$$.

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 $$$\sqrt{{{{x}}^{{2}}}}=\pm\sqrt{{{81}}}$$$ or more simply $$${x}=\pm{9}$$$.

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 $$${2}{{x}}^{{2}}+{5}={0}$$$.

Subtract $$${5}$$$ from both sides: $$${2}{{x}}^{{2}}=-{5}$$$.

Divide both sides by $$${2}$$$: $$${{x}}^{{2}}=-\frac{{5}}{{2}}$$$.

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: $$${3}{{x}}^{{2}}-{7}{x}={0}$$$.

Let's rewrite equation in the following form: $$${3}{x}\cdot{x}-{7}{x}={0}$$$.

Now, rewrite equation, using distributive property of multiplication: $$${x}{\left({3}{x}-{7}\right)}={0}$$$.

When is product of numbers equals 0?

When at least one factor equals 0.

So, either $$${x}={0}$$$ or $$${3}{x}-{7}={0}$$$.

Second equation is linear, its root is $$${x}=\frac{{7}}{{3}}$$$.

Answer: $$${x}={0}$$$ and $$${x}=\frac{{7}}{{3}}$$$.

We can generalize result of the above example.

Fact: incomplete quadratic equation $$${a}{{x}}^{{2}}+{b}{x}={0}$$$ has two roots: $$${x}={0}$$$ and $$${x}=-\frac{{b}}{{a}}$$$.

Now, it is time to exercise.

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

Answer: $$$\frac{{5}}{{2}}$$$ and $$$-\frac{{5}}{{2}}$$$.

Exercise 2. Solve the following: $$${3}{{x}}^{{2}}={0}$$$.

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

Exercise 3. Solve $$${2}{{z}}^{{2}}+{7}={0}$$$.

Answer: no roots.

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

Answer: $$${0}$$$ and $$$-\frac{{7}}{{2}}$$$.

Exercise 5. Solve $$$-{3}{{z}}^{{2}}=-{7}$$$.

Answer: $$$\sqrt{{\frac{{7}}{{3}}}}$$$ and $$$-\sqrt{{\frac{{7}}{{3}}}}$$$.

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

Answer: $$${0}$$$ and $$$-\frac{{9}}{{4}}$$$.