You factorise the quadratic expression x²+ (a+b) x +ab by rewriting it as the product of two binomials (x+a) X (x+b). By letting (a+b)=c and (ab)=d, you can recognize the familiar form of the quadratic equation x²+ cx+d. Factoring is the process of reverse multiplication and is the simplest way to solve quadratic equations.
Factor Quadratic Equations of the Form ex² +cx +d, e=1
Use the equation x²-10x+24 as an example and factorise it as the product of two binomials.
Rewrite this equation as follows: x²-10x+24= (x ?)(x ?).
Fill in the missing terms of the binomials with the two integers a and b whose product is +24, the constant term of x²-10x+24, and whose sum is -10, the coefficient of the x term. Since (-6) X (-4) = +24 and (-6) + (-4) = -10, then the correct factors of +24 are -6 and -4. So the equation x²-10x+24 = (x-4) (x-6).
Check that the binomial factors are correct by multiplying them together and comparing to the quadratic expression of this example.
Factor Quadratic Equations of the Form ex² +cx +d, e>1
Use the equation 3x² +5x-2 as an example and find the binomial factors.
Factor the equation 3x² +5x-2 by breaking down the 5x term into the sum of two terms, ax and bx. You choose a and b so that they add up to 5 and when multiplied together give the same product as the product of the coefficients of the first and last term of the equation 3x² +5x-2. Since (6-1) =5 and (6) X (-1) = (3) X (-2) then 6 and -1 are the correct coefficients for the x term.
Rewrite the x coefficients as the sum of 6 and -1 to get: 3x² + (6-1) x -2.
Distribute the x to both 6 and -1 and get: 3x² + 6 x -x -2. Then factor by grouping: 3x(x+2) + (-1) (x+2) = (3x-1) (x +2). This is the final answer.
Check the answer by multiplying the binomials (3x-1) (x +2) and compare to the quadratic equation of this example.
You cannot factorize all quadratic equations. In these special cases, you have to complete the square or use the quadratic formula.