Quadratics can be solved by equating one side of the equation to zero and factorising. For a product to be equal to zero, one or both of the factors must be equal to zero. Therefore, the quadratic equation can be solved by equating each factor to zero and solving for the variable.
Quadratic equations that cannot be factorised can be solved by completing the square. Completing the square results in a square plus a term, which can be easily solved.
Another method to solve quadratics which cannot be factorised is by using the quadratic formula. Simply substitute the coefficients of the quadratic equation into the formula to find the solution(s).
For any quadratic equation in the form ax² + bx + c = 0, the discriminant, b² − 4ac, can determine the number of roots the equation has. When the discriminant is positive, the quadratic equation has two distinct real roots. When the discriminant is 0, the quadratic equation has one real root. When the discriminant is negative, the quadratic equation has no real roots. The roots of the quadratic equation can then be calculated using the quadratic formula.
If it is found that the discriminant of a quadratic equation is greater than or equal to zero, ie. the equation has two distinct or equal real roots, the roots can be found by factorising, the quadratic equation or completing the square.
The Remainder Theorem states that if a polynomial, f(x), is divided by (x − a), the remainder is f(a). The Factor Theorem is a special case of the Remainder Theorem; when f(a) = 0, (x − a) is a factor of the polynomial and a is a solution to the polynomial, and vice versa.