To factor quadratic equations, the general approach is as follows:
- Write the quadratic in standard form ax2+bx+cax^2+bx+cax2+bx+c.
- Find two numbers whose product is a×ca\times ca×c and whose sum is bbb.
- Use those two numbers to split the middle term bxbxbx into two terms.
- Group the terms into two pairs and factor out the common factor from each pair.
- Factor out the common binomial factor from the grouped pairs to obtain the factors.
For example, if factoring 2x2+7x+32x^2+7x+32x2+7x+3:
- a×c=2×3=6a\times c=2\times 3=6a×c=2×3=6
- Find two numbers that multiply to 6 and add to 7, which are 6 and 1.
- Rewrite 7x7x7x as 6x+x6x+x6x+x, so 2x2+6x+x+32x^2+6x+x+32x2+6x+x+3.
- Group: (2x2+6x)+(x+3)(2x^2+6x)+(x+3)(2x2+6x)+(x+3).
- Factor each: 2x(x+3)+1(x+3)2x(x+3)+1(x+3)2x(x+3)+1(x+3).
- Factor out common binomial: (2x+1)(x+3)(2x+1)(x+3)(2x+1)(x+3).
Also, if the quadratic starts with a leading coefficient other than 1, methods like factoring out the greatest common factor first or using the "magic X" method can be helpful. If factoring is difficult or not obvious, the quadratic formula can find the roots and express the quadratic as a product of binomials based on those roots. This method works consistently for most quadratics and is a key skill in algebra.
