The book classifies quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones.
Historian Carl Boyer notes the following regarding the lack of modern abstract notations in the book:
... the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation (see History of algebra) found in the Greek Arithmetica or in Brahmagupta's work.
Even the numbers were written out in words rather than symbols!

— Carl B. Boyer, A History of Mathematics
Thus the equations are verbally described in terms of "squares" (what would today be "x²"), "roots" (what would today be "x") and "numbers" ("constants": ordinary spelled out numbers, like 'forty-two').
The six types, with modern notations, are:
squares equal roots (ax² = bx)
squares equal number (ax² = c)
roots equal number (bx = c)
squares and roots equal number (ax² + bx = c)
squares and number equal roots (ax² + c = bx)
roots and number equal squares (bx + c = ax²)
Islamic mathematicians, unlike the Hindus, did not deal with negative numbers at all; hence an equation like bx + c = 0 does not appear in the classification, because it has no positive solutions if all the coefficients are positive.
Similarly equation types 4, 5 and 6, which look equivalent to the modern eye, were distinguished because the coefficients must all be positive.

The al-ğabr ("forcing", "restoring") operation is moving a deficient quantity from one side of the equation to the other side.
In an al-Khwarizmi's example (in modern notation), "x² = 40x − 4x²" is transformed by al-ğabr into "5x² = 40x".
Repeated application of this rule eliminates negative quantities from calculations.

Al-Muqabala (المقابله, "balancing" or "corresponding") means subtraction of the same positive quantity from both sides:  "x² + 5 = 40x + 4x²" is turned into "5 = 40x + 3x²".
Repeated application of this rule makes quantities of each type ("square"/"root"/"number") appear in the equation at most once, which helps to see that there are only 6 basic solvable types of the problem, when restricted to positive coefficients and solutions.

Subsequent parts of the book do not rely on solving quadratic equations.
The book classifies quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones. Historian Carl Boyer notes the following regarding the lack of modern abstract notations in the book:
... the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation (see History of algebra) found in the Greek Arithmetica or in Brahmagupta's work. Even the numbers were written out in words rather than symbols! 
— Carl B. Boyer, A History of Mathematics
Thus the equations are verbally described in terms of "squares" (what would today be "x²"), "roots" (what would today be "x") and "numbers" ("constants": ordinary spelled out numbers, like 'forty-two'). The six types, with modern notations, are:
squares equal roots (ax² = bx)
squares equal number (ax² = c)
roots equal number (bx = c)
squares and roots equal number (ax² + bx = c)
squares and number equal roots (ax² + c = bx)
roots and number equal squares (bx + c = ax²)
Islamic mathematicians, unlike the Hindus, did not deal with negative numbers at all; hence an equation like bx + c = 0 does not appear in the classification, because it has no positive solutions if all the coefficients are positive. Similarly equation types 4, 5 and 6, which look equivalent to the modern eye, were distinguished because the coefficients must all be positive.
The al-ğabr ("forcing", "restoring") operation is moving a deficient quantity from one side of the equation to the other side. In an al-Khwarizmi's example (in modern notation), "x² = 40x − 4x²" is transformed by al-ğabr into "5x² = 40x". Repeated application of this rule eliminates negative quantities from calculations.
Al-Muqabala (المقابله, "balancing" or "corresponding") means subtraction of the same positive quantity from both sides:  "x² + 5 = 40x + 4x²" is turned into "5 = 40x + 3x²". Repeated application of this rule makes quantities of each type ("square"/"root"/"number") appear in the equation at most once, which helps to see that there are only 6 basic solvable types of the problem, when restricted to positive coefficients and solutions. 
Subsequent parts of the book do not rely on solving quadratic equations.
The Compendious Book on Calculation by Completion and Balancing