Many students get dazed and confused when confronted with quadratic equations. For most, it’s the appearance of the square exponent that gets them all riled up. However, Teacher Wacks Cagampan, senior math lecturer at Ahead Tutorial & Review Center, says it doesn’t have to be so. Here, he explains how to make heads and tails of these tricky equations.
First off, a quadratic equation is a second-order polynomial equation in a single variable. The word `quadratic’ comes from the Latin prefix `quad,’ meaning four, which is the number of sides in a square; thus, the 2 or square exponent.
Here’s a simple quadratic equation:
Y 2 (square exponent) = 4
Here, you are looking for a number (or Y) which, when multiplied by itself, will yield four. Since 2 when multiplied by itself (2 x 2) equals 4, then Y is 2. Y can also be -2 because -2 multiplied by itself (-2 x -2) equals 4.
Here’s another quadratic equation:
Y 2 (square exponent) = 49
Think: What number, when multiplied by itself, will yield 49? It’s not 3, because 3 multiplied by 3 equals 9. It’s not 6, because 6 multiplied by 6 equals 36. How about 7?
7 multiplied by 7 equals 49. That’s your Y!
Your Y can also be -7, because -7 multiplied by itself equals 49.
Here’s something a bit more complicated:
X2 (square exponent) + 3X + 2 = 0
Here’s a clue from Teacher Wacks: Find a number that, when used to replace all X’s, will yield zero through trial and error. Start with very small numbers like -1.
Substituting -1 for X in the above quadratic equation, we get:
-1 2 (square exponent) + 3 (-1) + 2 = 0
Let’s break down the elements of the equation one by one:
-1 2 (square exponent) = 1 (That’s because -1 multiplied by -1 equals 1)
3 multiplied by -1 equals -3
Now, let’s put them all together:
1 + -3 + 2 equals 0
X is thus -1!
Because quadratic equations involve splitting a number into two parts, or taking it apart, says AHEAD Tutorial & Review Center Teacher Wacks, it can easily lead us to another math concept, factoring. The idea is to break an equation into parts or factors as a systematic alternative to pure trial and error.
“A quadratic equation always has an exponent of 2. Thus, it always has two factors. If a quadratic equation is a parent, it will always have two -children’,” explains Teacher Wacks.
Your goal then is to find out how these children look like and how to put them together.
Let’s go back to X2 (square exponent) + 3X + 2 = 0.
Like all quadratic equations, it has two factors. To find out what they are, first put an x at the start of the two factors. Then put the plus sign because the operation requires addition.
It will look like this: (x + __ ) (x + __ ). Since the answer should be zero, you should make it (x+__ ) (x+__ ) = 0.
“Notice the pattern,” says Teacher Wacks.
“The third number in the equation is 2. Look for two numbers that when multiplied is equal to 2. Since 2 multiplied by 1 equals 2, the two numbers are 2 and 1. At the same time, when you add these two numbers, the result should be equal to the middle number. In this case, we are correct since 2 + 1 = 3, which the middle number in the original equation. Now put 2 and 1 in the missing spaces of the two factors. The result is (x +2) (x +1) = 0.”
For the final step, Teacher Wacks says that to find X, you must remember that the product in this equation is 0. Thus, the two factors, when multiplied, must amount to zero. Since (-2 + 2) = 0 and (-1 + 1) = 0, and multiplying (0) by (0) = 0, the answer is x = -2 or x = -1.