What is Quadratic Equation

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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`.