# What is Quadratic Equation

**Quadratic equation in one variable** is the equation with standard form $$$\color{purple}{{{a}{{x}}^{{2}}+{b}{x}+{c}={0}}}$$$.

$$$a$$$, $$$b$$$ and $$$c$$$ are some numbers and $$$x$$$ is variable. Note, that $$$a$$$ can't be zero.

In essence, quadratic equation is nothing more than quadratic polynomial ("quad" means square) on the left hand side, and zero on the right hand side.

Examples of quadratic equations are:

- $$${4}{{x}}^{{2}}-{2}{x}+{5}={0}$$$ ($$${a}={4}$$$, $$${b}=-{2}$$$, $$${c}={5}$$$)
- $$${{m}}^{{2}}-{1}={0}$$$ ($$${a}={1}$$$, $$${b}={0}$$$, $$${c}=-{1}$$$)
- $$$\frac{{3}}{{4}}{{y}}^{{2}}-{3}{y}={0}$$$ ($$${a}=\frac{{3}}{{4}}$$$, $$${b}=-{3}$$$, $$${c}={0}$$$)

Using equivalence of equations and properties of expressions, we can convert some other equations into the standard form:

- $$$-{2}{{x}}^{{2}}={1}+{3}{x}$$$ is equivalent to $$$-{2}{{x}}^{{2}}-{3}{x}-{1}$$$ (move everything to the left)
- $$${2}{\left({{x}}^{{2}}-{5}{x}\right)}={4}$$$ becomes $$${2}{{x}}^{{2}}-{10}{x}-{4}={0}$$$ (use distributive property of multiplication to expand left hand side, then move $$${4}$$$ to the left)
- $$${y}{\left({2}-{y}\right)}={4}{y}+{3}$$$ becomes $$$-{{y}}^{{2}}-{2}{y}-{3}={0}$$$ (multiply monomial by polynomial, move everything to the left and combine like terms)
- $$${\left({x}-{4}\right)}{\left({x}+{5}\right)}={1}$$$ becomes $$${{x}}^{{2}}+{x}-{21}={0}$$$ (multiply binomials, then move $$${1}$$$ to the left)
- $$${x}+\frac{{1}}{{x}}={3}$$$ is equivalent to $$${{x}}^{{2}}-{3}{x}+{1}={0}$$$ (multiply both sides by $$${x}$$$, then move everything to the left)

**Equation is quadratic**, when it is written in standard form and variable is raised to the second power and to the first power only.

Following are **NOT** linear equations:

- $$${2}{{x}}^{{3}}+{3}={0}$$$ (variable raised to the third power)
- $$${2}{y}-{3}=\frac{{3}}{{2}}{{y}}^{{4}}$$$ (there is variable, raised to the fourth power)
- $$$\frac{{1}}{{y}}+{{y}}^{{2}}={2}$$$ (if we multiply both sides by $$${y}$$$, we get the following equation: $$${1}+{{y}}^{{3}}={2}{y}$$$ and this equation is not quadratic)

**Exercise 1.** Determine, whether $$${2}{{x}}^{{2}}=-{5}{x}+{3}$$$ is quadratic and write it in standard form if it is.

**Answer**: yes; $$${2}{{x}}^{{2}}+{5}{x}-{3}={0}$$$.

**Exercise 2.** Determine, whether $$${x}{\left({x}-{2}\right)}={x}$$$ is quadratic and write it in standard form if it is.

**Answer**: yes; $$${{x}}^{{2}}-{3}{x}={0}$$$.

**Exercise 3.** Determine, whether $$${{x}}^{{3}}={7}$$$ is quadratic and write it in standard form if it is.

**Answer**: no.

**Exercise 4.** Determine, whether $$$\frac{{1}}{{x}}+{5}={{x}}^{{2}}$$$ is quadratic and write it in standard form if it is.

**Answer**: no. Multiplying both sides by $$${x}$$$ gives $$${1}+{5}{x}={{x}}^{{3}}$$$.

**Exercise 5.** Determine, whether $$$\frac{{3}}{{x}}={x}+{4}$$$ is quadratic and write it in standard form.

**Answer**: yes; $$$-{{x}}^{{2}}-{4}{x}+{3}={0}$$$.