## Related calculator: Quadratic Equation Calculator

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.

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 ${2}{{z}}^{{2}}+{7}={0}$.

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 $-\frac{{9}}{{4}}$.