
Here we will show you how to factor the quadratic function -86x² - 96x - 10 using the box method. In other words, we will show you how to factor negative 86x squared minus 96x minus 10 (-86x^2 - 96x - 10) 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 86x² + 96x + 10. Now we can label the different parts of our equation, like this:
a = 86
b = 96
c = 10
Step 2: Next, we need to draw a box and divide it into four squares:
10 | 10x | 10 |
86x | 86x² | 86x |
x | 1 |
10 | 10x | 10 |
86x | 86x² | 86x |
x | 1 |
More specifically, 86 times 10 is 860. Therefore, we need to find the two numbers that multiply to equal 860, and add to equal 96.
? × ? = 860
? + ? = 96
After looking at this problem, we can see that the two numbers that multiply together to equal 860, and add together to equal 96, are 10 and 86, as illustrated here:
10 × 86 = 860
10 + 86 = 96
Now, we can fill in the last two squares in our box with 10x and 86x. Place 10x in the upper left square, and place 86x in the lower right square.
10 | 10x | 10 |
86x | 86x² | 86x |
x | 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 10x and 10. The greatest common factor of 10x and 10 is 10. Therefore, we write 10 to the left of the top row. You can see it here in the color green:
10 | 10x | 10 |
86x | 86x² | 86x |
x | 1 |
10 | 10x | 10 |
86x | 86x² | 86x |
x | 1 |
86x² ÷ 86x = x
You can see this value colored in orange below:
10 | 10x | 10 |
86x | 86x² | 86x |
x | 1 |
Next, we divide 86x by 86x (labeled in blue). This gives us 1.
86x ÷ 86x = 1
You can see this value colored in purple below:
10 | 10x | 10 |
86x | 86x² | 86x |
x | 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 -86x² - 96x - 10. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:
(86x + 10)(x + 1)
In our original quadratic equation, -86x² - 96x - 10, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:
-(86x + 10)(x + 1)
That’s it! Now you know how to factor the equation -86x² - 96x - 10.
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