Here we will show you how to factor the quadratic function 14x² + 73x + 78 using the box method. In other words, we will show you how to factor 14x squared plus 73x plus 78 (14x^2 + 73x + 78) using the box method. It is a 5-step process:
Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 14x² + 73x + 78, like this:
a = 14
b = 73
c = 78
Step 2: Next, we need to draw a box and divide it into four squares:
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
More specifically, 14 times 78 is 1092. Therefore, we need to find the two numbers that multiply to equal 1092, and add to equal 73.
? × ? = 1092
? + ? = 73
After looking at this problem, we can see that the two numbers that multiply together to equal 1092, and add together to equal 73, are 21 and 52, as illustrated here:
21 × 52 = 1092
21 + 52 = 73
Now, we can fill in the last two squares in our box with 21x and 52x. Place 21x in the upper left square, and place 52x in the lower right square.
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
Our goal is to find the numbers that, when multiplied together, result in the products in each square. We can do this by finding the greatest common factor in each row.
Let’s look at the top row. We have the terms 21x and 78. The greatest common factor of 21x and 78 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
14x² ÷ 2x = 7x
You can see this value colored in orange below:
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
Next, we divide 52x by 2x (labeled in blue). This gives us 26.
52x ÷ 2x = 26
You can see this value colored in purple below:
3 | 21x | 78 |
2x | 14x² | 52x |
7x | 26 |
Step 5: Now we have all of the information we need to finish factoring the equation. The values outside of the box are used to factor 14x² + 73x + 78. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:
(2x + 3)(7x + 26)
That’s it! Now you know how to factor the equation 14x² + 73x + 78.
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