randRange( 1, 4 ) / randRangeNonZero( -2, 2 ) ( randRange( -3, 3 ) * 2 + 1 ) / 2 ( X1 + X2 ) * -1 B > 0 ? "+" : "-" X1 * X2 getLCM( toFraction( B )[1], toFraction( C )[1] ) new Polynomial( 0, 2, [MULT*C, MULT*B, MULT], "x" ) POLY.text() $._("or")

Complete the square to solve for `x`.

`POLY_TEXT = 0`

B / 2 C * -1 + pow( B / 2, 2 ) X1 X2
B / 2 C * -1 + pow( B / 2, 2 ) X2 X1
Completed Square:
`(x + {}` `)^2 = {}`
Solution:
`x = {}``\quad\text{OR}\quad x = {}`
( randRange( -4, 4 ) * 2 + 1 ) / 2 X1

Completed Square:
`(x + {}`-X1 `)^2 = {}` 0
Solution:
`x = \quad` X1

randRangeNonZero( -8, 8 ) randRange( -4, 4 ) * 2 + ( X1 % 2 - 1 )

First, divide the polynomial by `MULT`, the coefficient of the `x^2` term.

`x^2 + decimalFraction( B, 1, 1 )B_SIGNx + decimalFraction( C, 1, 1 ) = 0`

Move the constant term to the right side of the equation.

`x^2 + decimalFraction( B, 1, 1 )B_SIGNx = decimalFraction( C * -1, 1, 1 )`

We complete the square by taking half of the coefficient of our `x` term, squaring it, and adding it to both sides of the equation. The coefficient of our `x` term is `decimalFraction( B, 1, 1 )`, so half of it would be `decimalFraction( B / 2, 1, 1 )`, and squaring it gives us `\color{blue}{decimalFraction( pow( B / 2, 2 ), 1, 1 )}`.

`x^2 + decimalFraction( B, 1, 1 )B_SIGNx \color{blue}{ + decimalFraction( pow( B / 2, 2 ), 1, 1 )} = decimalFraction( C * -1, 1, 1 ) \color{blue}{ + decimalFraction( pow( B / 2, 2 ), 1, 1 )}`

We can now rewrite the left side of the equation as a squared term.

`( x + decimalFraction( B / 2, 1, 1 ) )^2 = decimalFraction( C * -1 + pow( B / 2, 2 ), 1, 1 )`

Note that the left side of the equation is already a perfect square trinomial. The coefficient of our `x` term is `decimalFraction( B, 1, 1 )`, half of it is `decimalFraction( B / 2, 1, 1 )`, and squaring it gives us `\color{blue}{decimalFraction( pow( B / 2, 2 ), 1, 1 )}`, our constant term.

Thus, we can rewrite the left side of the equation as a squared term.

`( x + decimalFraction( B / 2, 1, 1 ) )^2 = decimalFraction( C * -1 + pow( B / 2, 2 ), 1, 1 )`

Take the square root of both sides.

`x + decimalFraction( B / 2, 1, 1 ) = \pmdecimalFraction( sqrt( C * -1 + pow( B / 2, 2 ) ), 1, 1 )`

Isolate `x` to find the solution(s).

`x = decimalFraction( -B / 2, 1, 1 )\pmdecimalFraction( sqrt( C * -1 + pow( B / 2, 2 ) ), 1, 1 )`

The solutions are: `x = decimalFraction( -B / 2 + sqrt( C * -1 + pow( B / 2, 2 ) ), 1, 1 ) \text{ OR } x = decimalFraction( -B / 2 - sqrt( C * -1 + pow( B / 2, 2 ) ), 1, 1 )`

The solution is: `x = decimalFraction( -B / 2 + sqrt( C * -1 + pow( B / 2, 2 ) ), 1, 1 )`

We already found the completed square: `( x + decimalFraction( B / 2, 1, 1 ) )^2 = decimalFraction( C * -1 + pow( B / 2, 2 ), 1, 1 )`