Factor -48x² - 89x

Factor -48x² - 89x - 38

Here we will show you how to factor the quadratic function -48x² - 89x - 38 using the box method. In other words, we will show you how to factor negative 48x squared minus 89x minus 38 (-48x^2 - 89x - 38) 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 48x² + 89x + 38. Now we can label the different parts of our equation, like this:

a = 48
b = 89
c = 38

Step 2: Next, we need to draw a box and divide it into four squares:

2	32x	38
3x	48x²	57x
	16x	19

We put 48x² (a) in the bottom left square and 38 (c) in the top right square, like this:

2	32x	38
3x	48x²	57x
	16x	19

Step 3: In order to fill in the other two squares, we need to do some math. We have to find two numbers that multiply together to give us the product of 48 times 38 (a × c), and add together to equal 89 (b).

More specifically, 48 times 38 is 1824. Therefore, we need to find the two numbers that multiply to equal 1824, and add to equal 89.

? × ? = 1824
? + ? = 89

After looking at this problem, we can see that the two numbers that multiply together to equal 1824, and add together to equal 89, are 32 and 57, as illustrated here:

32 × 57 = 1824
32 + 57 = 89

Now, we can fill in the last two squares in our box with 32x and 57x. Place 32x in the upper left square, and place 57x in the lower right square.

2	32x	38
3x	48x²	57x
	16x	19

Step 4: Next, we need to find four final numbers to finish factoring our equation. We can see that our box is a 2 x 2 table made up of rows and columns.

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 32x and 38. The greatest common factor of 32x and 38 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	32x	38
3x	48x²	57x
	16x	19

Next, let’s look at the bottom row. We have the terms 48x² and 57x. The greatest common factor of 48x² and 57x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

2	32x	38
3x	48x²	57x
	16x	19

To find the values below the table, we first divide 48x² by 3x (labeled in blue). This gives us 16x.

48x² ÷ 3x = 16x

You can see this value colored in orange below:

2	32x	38
3x	48x²	57x
	16x	19

Next, we divide 57x by 3x (labeled in blue). This gives us 19.

57x ÷ 3x = 19

You can see this value colored in purple below:

2	32x	38
3x	48x²	57x
	16x	19

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 -48x² - 89x - 38. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(3x + 2)(16x + 19)

In our original quadratic equation, -48x² - 89x - 38, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x + 2)(16x + 19)

That’s it! Now you know how to factor the equation -48x² - 89x - 38.

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