Non-Quadratic Binomials

• 1). Move all nonzero terms to one side of the equation, via addition or subtraction, so that the equation equals zero. For example, in -12c = -8c^2, add -8c^2 to both "c" sides, obtaining -12c + 8c^2 = 0
  
  
• 2). Write the terms in descending order, so that the term with the greatest exponent appears furthest to the left, followed by the term with the second-greatest exponent, and so on. The equation -12c + 8c^2 = 0 becomes 8c^2 -- 12c = 0.
  
  
• 3). Determine the GCF, or greatest common factor, which is the largest number and highest degree variable that factors into all terms in the polynomial. In 8c^2 -- 12c = 0, the GCF is 4c.
  
  
• 4). Factor the GCF out of the equation. Draw a set of parentheses and put the GCF to their left. Inside the parentheses, place the terms that can be multiplied by the GCF to produce the equation from Step 3. It may be helpful to think of this as reversing the distributive property. Factoring the GCF out of 8c^2 -- 12c = 0 yields 4c (2c -- 3) = 0.
  
  
• 5). Split the equation into two separate mini-equations. Set the term outside the parentheses equal to zero and set the terms inside the parentheses equal to zero. The equation 4c (2c -- 3) = 0 is split into 4c = 0 and 2c -- 3 = 0.
  
  
• 6). Solve both equations for the given variable. Solving 4c = 0 produces c = 0, and solving 2c -- 3 = 0 produces c = 3/2, or 1.5. These are your solutions.
  
  

Quadratic Polynomials

• 1). Use the quadratic formula. Move all nonzero terms to the same side of the equation and rewrite them in descending order as described in Section 1. For example, the equation 8 + 3x^2 = -10x becomes 3x^2 + 10x + 8.
  
  
• 2). Identify "a," "b" and "c." Quadratic polynomials take the form ax^2 + bx + c, in which "x" is a variable and "a," "b" and "c" are numbers. In 3x^2 + 10x + 8, a = 3, b = 10 and c = 8.
  
  
• 3). Insert the values for "a," "b" and "c" into the quadratic formula. The quadratic formula is: --b plus or minus the square root of the quantity b^2 -- 4ac, all divided by 2a. In the example, write: -10 plus or minus the square root of the quantity 10^2 -- 4(3)(8) divided by 2(3).
  
  
• 4). Simplify the portion inside the square root by following the order of operations, commonly known as PEMDAS. In the example, 10^2 -- 4(3)(8) becomes 100 -- 96, which simplifies to 4. Take the square root of this, obtaining 2 in the example. Now the formula is simplified to: -10 plus or minus 2, divided by 2(3).
  
  
• 5). Simplify the denominator. In the example, multiply 2 by 3 to get 6.
  
  
• 6). Separate into two problems. One will contain a plus sign following the -b, and the other will contain a minus sign. In the example, your two expressions are: (-10 + 2) / 6 and (-10 -- 2) / 6.
  
  
• 7). Simplify each expression by first performing the addition or subtraction and then performing the division. In the example, you get -8/6, which simplifies to -4/3, and -12/6, which simplifies to -2. These are your answers; write them with a variable and equals sign. The example produces x = -4/3 and x = -2.