2x2−8x−4=3x−x2{\displaystyle 2x^{2}-8x-4=3x-x^{2}} 2x2+x2−8x−3x−4=0{\displaystyle 2x^{2}+x^{2}-8x-3x-4=0} 3x2−11x−4=0{\displaystyle 3x^{2}-11x-4=0}
Since 3x2{\displaystyle 3x^{2}} only has one set of possible factors, 3x{\displaystyle 3x} and x{\displaystyle x}, you can write those in the parenthesis: (3x±?)(x±?)=0{\displaystyle (3x\pm ?)(x\pm ?)=0}. Then, use the process of elimination to plug in the factors of 4 to find a combination that produces -11x when multiplied. You can either use a combination of 4 and 1, or 2 and 2, since both of those numbers multiply to get 4. Just remember that one of the terms should be negative, since the term is -4. [3] X Research source By trial and error, try out this combination of factors (3x+1)(x−4){\displaystyle (3x+1)(x-4)}. When you multiply them out, you get 3x2−12x+x−4{\displaystyle 3x^{2}-12x+x-4}. If you combine the terms −12x{\displaystyle -12x} and x{\displaystyle x}, you get −11x{\displaystyle -11x}, which is the middle term you were aiming for. You have just factored the quadratic equation. As an example of trial and error, let’s try checking a factoring combination for 3x2−11x−4=0{\displaystyle 3x^{2}-11x-4=0} that is an error (does not work): (3x−2)(x+2){\displaystyle (3x-2)(x+2)} = 3x2+6x−2x−4{\displaystyle 3x^{2}+6x-2x-4}. If you combine those terms, you get 3x2−4x−4{\displaystyle 3x^{2}-4x-4}. Though the factors -2 and 2 do multiply to make -4, the middle term does not work, because you needed to get −11x{\displaystyle -11x}, not −4x{\displaystyle -4x}.
Solve 3x + 1 = 0 3x = -1 . . . . . by subtracting 3x/3 = -1/3 . . . . . by dividing x = -1/3 . . . . . simplified Solve x - 4 = 0 x = 4 . . . . . by subtracting x = (-1/3, 4) . . . . . by making a set of possible, separate solutions, meaning x = -1/3, or x = 4 seem good.
So, both solutions do “check” separately, and both are verified as working and correct for two different solutions.
4x2 - 5x - 13 = x2 -5 4x2 - x2 - 5x - 13 +5 = 0 3x2 - 5x - 8 = 0
{-b +/-√ (b2 - 4ac)}/2 {-(-5) +/-√ ((-5)2 - 4(3)(-8))}/2(3) = {-(-5) +/-√ ((-5)2 - (-96))}/2(3)
{-(-5) +/-√ ((-5)2 - (-96))}/2(3) = {5 +/-√(25 + 96)}/6 {5 +/-√(121)}/6
(5 + 11)/6 (5 - 11)/6
(5 + 11)/6 = 16/6 (5-11)/6 = -6/6
16/6 = 8/3 -6/6 = -1 x = (-1, 8/3)
2x2 - 9 = 12x = 2x2 - 12x - 9 = 0 In this equation, the a term is 2, the b term is -12, and the c term is -9.
2x2 - 12x - 9 = 0 2x2 - 12x = 9
2x2/2 - 12x/2 = 9/2 = x2 - 6x = 9/2
-6/2 = -3 = (-3)2 = 9 = x2 - 6x + 9 = 9/2 + 9
x = 3 + 3(√6)/2 x = 3 - 3(√6)/2)
