Here we will show you how to factor the quadratic function -92x² - 78x + 62 using the box method. In other words, we will show you how to factor negative 92x squared minus 78x plus 62 (-92x^2 - 78x + 62) 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 + 78x - 62. Now we can label the different parts of our equation, like this:
a = 92
b = 78
c = -62
Step 2: Next, we need to draw a box and divide it into four squares:
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
More specifically, 92 times -62 is -5704. Therefore, we need to find the two numbers that multiply to equal -5704, and add to equal 78.
? × ? = -5704
? + ? = 78
After looking at this problem, we can see that the two numbers that multiply together to equal -5704, and add together to equal 78, are -46 and 124, as illustrated here:
-46 × 124 = -5704
-46 + 124 = 78
Now, we can fill in the last two squares in our box with -46x and 124x. Place -46x in the upper left square, and place 124x in the lower right square.
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
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 -46x and -62. The greatest common factor of -46x and -62 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
92x² ÷ 4x = 23x
You can see this value colored in orange below:
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
Next, we divide 124x by 4x (labeled in blue). This gives us 31.
124x ÷ 4x = 31
You can see this value colored in purple below:
-2 | -46x | -62 |
4x | 92x² | 124x |
23x | 31 |
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 - 78x + 62. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:
(4x - 2)(23x + 31)
In our original quadratic equation, -92x² - 78x + 62, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:
-(4x - 2)(23x + 31)
That’s it! Now you know how to factor the equation -92x² - 78x + 62.
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