Here we will show you how to factor the quadratic function -92x² + 33x + 14 using the box method. In other words, we will show you how to factor negative 92x squared plus 33x plus 14 (-92x^2 + 33x + 14) using the box method. It is a 5-step process:
Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 92x² - 33x - 14. Now we can label the different parts of our equation, like this:
a = 92
b = -33
c = -14
Step 2: Next, we need to draw a box and divide it into four squares:
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
More specifically, 92 times -14 is -1288. Therefore, we need to find the two numbers that multiply to equal -1288, and add to equal -33.
? × ? = -1288
? + ? = -33
After looking at this problem, we can see that the two numbers that multiply together to equal -1288, and add together to equal -33, are -56 and 23, as illustrated here:
-56 × 23 = -1288
-56 + 23 = -33
Now, we can fill in the last two squares in our box with -56x and 23x. Place -56x in the upper left square, and place 23x in the lower right square.
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
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 -56x and -14. The greatest common factor of -56x and -14 is -14. Therefore, we write -14 to the left of the top row. You can see it here in the color green:
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
92x² ÷ 23x = 4x
You can see this value colored in orange below:
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
Next, we divide 23x by 23x (labeled in blue). This gives us 1.
23x ÷ 23x = 1
You can see this value colored in purple below:
-14 | -56x | -14 |
23x | 92x² | 23x |
4x | 1 |
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 -92x² + 33x + 14. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:
(23x - 14)(4x + 1)
In our original quadratic equation, -92x² + 33x + 14, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:
-(23x - 14)(4x + 1)
That’s it! Now you know how to factor the equation -92x² + 33x + 14.
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