Understanding the quadratic equation can be confusing and difficult for the beginner. With so many terms and possibilities, the frustration often is overwhelming. This usually leads to the problem solver to giving up. But with any mathematical problem, it can easily be solved by breaking it down into a series of steps.
For the purposes of this article so of the symbols are changed to fit within the format. The symbols are: + for addition, – for subtraction, * for multiplication, / for division, ^ means raised to a power, and x is unknown.
Step 1: Recognize x is an unknown number.
The x is often represented as an unknown number. It is treated exactly like a number.
For example, x + x = 2x, so if x = 3 then 3 + 3 = 2 * 3 or 3 + 3 = 6.
For example, x – x = 0, so if x = 1 then 1 – 1 = 0.
For example, x * x = x^2, so if x = 2 then 2 * 2 = 2^2 or 2 * 2 = 4.
For example, x/x = 1, so if x = 5 then 5/5 = 1.
Step 2: What happens to one side of the equation, must happen to the other side.
The two sides of an equation are equal. So treat both sides the same way.
For example, 2x + 3 = 6 + x. In order to solve for x, we must get x on one side of the equation.
If I want to get x to the right side of the equation, I would subtract an x from both sides.
2x + 3 = 6 + x
-x = – x
–
x + 3 = 6
This puts the x on side, making the equation easier to solve.
x + 3 = 6
– 3 = – 3
–
x = 3
The easiest way to check if this is the correct answer is to put it back in the equation.
2x + 3 = 6 + x and x = 3 then 2 * 3 + 3 = 6 + 3, 6 + 3 = 6 + 3, 9 = 9. This is correct.
If for some reason, a mistake was made and it came out as x = 1 then
2 * 1 + 3 = 6 + 1, 2 + 3 = 6 + 1, 5 = 7. This is incorrect. Go back and check the calculations.
Another example, 4x – 2 + x = x – 4 + 2x. In order to solve for x in this equation, the easiest thing to in this case is to combine like terms. 4x + x = 5x and x + 2x = 3x. Rewrite the equation like this 5x – 2 = 3x – 4. Now to put x on the right side of the equation:
5x – 2 = 3x – 4
– 3x = -3x
–
2x – 2 = – 4
To get x all by itself:
2x – 2 = – 4
+ 2 = + 2
–
2x = – 2
2x/2 = – 2/2
x = – 1
To check put the answer back in the equation 4 * (-1) -2 + (-1) = (-1) – 4 + 2 * (-1)
– 7 = – 7 so this answer is correct.
Step 3: Recognizing the Quadratic equation.
The form for the quadratic equation is x^2 + yx + z = 0. Note y and z are only used in place to represent a number. This is the form that the equation should take before it can be solved.
For example: 2x^2 + 3x – 4 = x^2 + 8 – x. The left side of the equation should equal zero so,
2x^2 + 3x – 4 = x^2 + 8 – x
– x^2 = -x^2
–
x^2 + 3x – 4 = 0 + 8 – x
Now: x^2 + 3x – 4 = 8 – x
+ x = 8 + x
–
x^2 + 4x – 4 = 8
Then: x^2 + 4x – 4 = 8
– 8 = – 8
–
x^2 + 4x – 12 = 0
This is the proper format for the quadratic equation. Noted both sides of the equation were treated the exact same way.
Step 4: Write all the factors for the first and last term in the equation.
In the equation above: x^2 + 4x – 12 = 0.
The factors for x^2 is (x,x)
The factors for – 12 is ( -1,12): (1, -12): (-2,6): (2, -6): (-3,4): (3,- 4)
Step 5: Place the first factors in the first position. Because there is only one set of factors for this example, the first factor gets placed in the front.
(x )(x ) = 0
Step 6: Find the factors that equal the middle number when added or subtracted.
-1 + 12 = 11 so this does not work. 1 – 12 = – 11 this does not work.
-2 + 6 = 4. Yeah a factor that matches the middle number.
2 – 6 = – 4 this does not work because the middle number is positive not negative.
-3 + 4 = – 1 so this does not work. 3 – 4 = – 1 so this does not work.
Step 7: Place the correct factors in the brackets.
(x – 2)(x + 6) = 0
Step 8: Double check.
To ensure that the problem is solved correctly multiply the first terms together
(x – 2)(x + 6): x * x = x^2 this matches the first part of the equation x^2 + 4x – 12 = 0.
Multiply the last terms together: -2 * 6 = – 12 which matches the last number in the equation.
Finally, and this is the tricky part, multiply the first term in the first bracket with the second term in the second bracket: (x )( + 6): x * 6 = 6x
Then multiply the second term in the first bracket with the first term in the second bracket:
( – 2)(x ): -2 * x = -2x
It should look like this when put together: x^2 + 6x – 2x -12 = 0 when simplified it is:
x^2 + 4x – 12 = 0. Which matches the original equation.
Step 9: Set the terms equal to zero.
For example: (x -2) = 0 or (x + 6) = 0.
Step 10: Solve for x
x – 2 = 0 so x = 2
x + 6 = 0 so x = – 6
Step 11: Write the answer.
(2,-6) In the equation: x^2 + 4x – 12 = 0: x = 2, x = -6
Breaking down and taking a quadratic equation step by step is the key to solving them. The answer can be easily checked merely by putting the numbers back in the equation. Take it one step at a time and with a little practice you can solve them easily.
