# What is Quadratic Equation

## Related calculator: Quadratic Equation Calculator

**Quadratic equation in one variable** is the equation with standard form `color(purple)(ax^2+bx+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:

- `4x^2-2x+5=0` (`a=4` , `b=-2`, `c=5`)
- `m^2-1=0` (`a=1` , `b=0`, `c=-1`)
- `3/4y^2-3y=0` (`a=3/4` , `b=-3`, `c=0`)

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

- `-2x^2=1+3x` is equivalent to `-2x^2-3x-1` (move everything to the left)
- `2(x^2-5x)=4` becomes `2x^2-10x-4=0` (use distributive property of multiplication to expand left hand side, then move `4` to the left)
- `y(2-y)=4y+3` becomes `-y^2-2y-3=0` (multiply monomial by polynomial, move everything to the left and combine like terms)
- `(x-4)(x+5)=1` becomes `x^2+x-21=0` (multiply binomials, then move `1` to the left)
- `x+1/x=3` is equivalent to `x^2-3x+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:

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

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

**Answer**: yes; `2x^2+5x-3=0`.

**Exercise 2.** Determine, whether `x(x-2)=x` is quadratic and write it in standard form if it is.

**Answer**: yes; `x^2-3x=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 `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+5x=x^3`.

**Exercise 5.** Determine, whether `3/x=x+4` is quadratic and write it in standard form.

**Answer**: yes; `-x^2-4x+3=0`.