How to Solve Quadratic Equations

Step-by-Step Guide

1. 1

   Step 1: Combine all of the like terms and move them to one side of the equation.

   The first step to factoring an equation is to move all of the terms to one side of the equation, keeping the x2{\displaystyle x^{2}} term positive.
   
   To combine the terms, add or subtract all of the x2{\displaystyle x^{2}} terms, the x{\displaystyle x} terms, and the constants (integer terms), moving them to one side of the equation so that nothing remains on the other side.
   
   Once the other side has no remaining terms, you can just write "0" on that side of the equal sign.
   
   Here's how you do it: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}
2. 2

   Step 2: Factor the expression.

   To factor the expression, you have to use the factors of the x2{\displaystyle x^{2}} term (3), and the factors of the constant term (-4), to make them multiply and then add up to the middle term, (-11).
   
   Here's how you do it:
   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 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.
   
   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}. , This will lead you to find two values for x{\displaystyle x} that will make the entire equation equal to zero, (3x+1)(x−4){\displaystyle (3x+1)(x-4)} =
   0.
   
   Now that you've factored the equation, all you have to do is put the expression in each set of parenthesis equal to zero.
   
   But why?
   -- because to get zero by multiplying, we have the "principle, rule or property" that one factor must be zero, then at least one of the factors in parentheses, as (3x+1)(x−4){\displaystyle (3x+1)(x-4)} must be zero; so, either (3x + 1) or else (x
   - 4) must equal zero.
   
   So, you would write 3x+1=0{\displaystyle 3x+1=0} and alsox−4=0{\displaystyle x-4=0}. , In a quadratic equation, there will be two possible values for x.
   
   Find x for each possible value of x one by one by isolating the variable and writing down the two solutions for x as the final solution.
   
   Here's how you do it:
   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. , 0 ..... by substituting (-1 + 1)(-4 1/3) ?=? 0 ..... by simplifying (0)(-4 1/3) = 0 ..... by multiplying therefore 0 = 0 .....
   
   Yes, x =
   -1/3 works , 0 ..... by substituting (13)(4 – 4) ?=? 0 ..... by simplifying (13)(0) = 0 ..... by multiplying 0 = 0 .....
   
   Yes, x = 4 works So, both solutions do "check" separately, and both are verified as working and correct for two different solutions.
3. 3

   Step 3: Set each set of parenthesis equal to zero as separate equations.

4. 4

   Step 4: Solve each "zeroed" equation independently.

5. 5

   Step 5: Check x = -1/3 in (3x + 1)(x – 4) = 0: We have (3+ 1)(– 4) ?=?

6. 6

   Step 6: Check x = 4 in (3x + 1)(x - 4) = 0: We have (3+ 1)(– 4) ?=?

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