Solving Quadratics

HOLLY DEROSA AND CHIARA CZULNO

Steps to Turning a Quadratic Equation into Standard Form :

Equation : 4x^2 - 2x = x^2 + 5
  • To solve this equation we must put it into standard form or ax^2+bx+c = 0.
  • To do that, we must first move x^2 to the left of the equal sign. Notice that 4x^2 and x^2 both have "^2" or are to the power of 2. Because of this, we can combine both numbers.
  • Because x^2 is a positive number, we're going to subtract it from 4x^2. Always do the opposite of what the number is. If x^2 was a negative number, we would be adding it.
  • 4x^2 -x^2 = 3x^2. When subtracting or adding numbers with powers, never subtract or add the powers. The ^2 stays the same.
  • Now our equation is 3x^2 - 2x = 5.
  • Since no other number to the left of the equal sign is similar to 5, we are just going to move it over. 5 is positive, so we'll make the 5 we move over negative. Our equation is now 3x^2 - 2x - 5 = 0.
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Steps to Solving/Factoring a Quadratic Formula :

  • Now we must factor the equation.
  • First, multiply 3x and -5, and get -15x-. Now, find two numbers that when multiplied equal -15, and when added equal -2. Those two numbers are -5, and 3.
  • Now, replace "-2x" with "-5x" and "3x". Our equation is now 3x^2 + 3x - 5x - 5 = 0.
  • Divide the equation in half, and find the greatest common factor or GCF from each half. 3x^2+3x and -5x-5.
  • The greatest common factor from 3x^2 + 3x is 3x, and the greatest common factor from -5x -5 is -5.
  • Now, divide the GCF from the first half by 3x. That leaves us with 1. Next, divide the GCF of the second half by -5. That leaves us 1.
  • I'm going to put the GCF outside of my parenthesis and put "x+1" inside my parenthesis.
  • 3x( x + 1 ) -5( x + 1)
  • Now, I'll take the GCFs and put them in their own parenthesis, and put x+1 in another parenthesis.
  • ( 3x - 5 ) ( x + 1)
  • In my second parenthesis, I am left with x + 1, or x + 1 = 0. I am first going to subtract 1 (remember to subtract if the number is positive, or add if the number is negative) from 1 and 0. That leaves me with x + 0 = -1, so the answer is x = -1.
  • In the first parenthesis, I have 3x - 5, or 3x - 5 = 0. First, add 5 to -5 and 0. That leaves me with 3x = 5. Divide 3x and 5 by 3, which leaves us with x = 5/3.
  • The answer to 4x^2 - 2x = x^2 + 5 is {-1, 5/3}

Questions Relating to Quadratic Equations :

1. What jobs use quadratic equations? Quadratic equations are obviously used in engineering, and jobs that fall in math and science categories, but are also used in jobs relating to insurance, because many insurance plans are based on these equations, as well as agriculture, because they may need this equation to find areas of field or calculate lengths and widths.

2. What is the first step in solving this equation : x^2 - 4x + 4 ?

The first step in solving this equation is to factor, or find two numbers that equal 4 when multiplied, and equal -4 when added.
3. How are quadratic equations used in everyday life?
Quadratic equations are used to find speed, figure out profit, find areas, finding velocity, etc.
4. What makes an equation quadratic?
An equation is quadratic when the first number and variable are squared, or to the second power. For example, 5x^2.


1. What are the characteristics of a quadratic equation?