Factor -48x² - 90x - 42


Factoring Quadratics

Here we will show you how to factor the quadratic function -48x² - 90x - 42 using the box method. In other words, we will show you how to factor negative 48x squared minus 90x minus 42 (-48x^2 - 90x - 42) 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 + 90x + 42. Now we can label the different parts of our equation, like this:

a = 48
b = 90
c = 42


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

42  42x 42
48x  48x² 48x
x 1
We put 48x² (a) in the bottom left square and 42 (c) in the top right square, like this:

42  42x 42
48x  48x² 48x
x 1
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 42 (a × c), and add together to equal 90 (b).

More specifically, 48 times 42 is 2016. Therefore, we need to find the two numbers that multiply to equal 2016, and add to equal 90.

? × ? = 2016
? + ? = 90

After looking at this problem, we can see that the two numbers that multiply together to equal 2016, and add together to equal 90, are 42 and 48, as illustrated here:

42 × 48 = 2016
42 + 48 = 90

Now, we can fill in the last two squares in our box with 42x and 48x. Place 42x in the upper left square, and place 48x in the lower right square.

42  42x 42
48x  48x² 48x
x 1
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 42x and 42. The greatest common factor of 42x and 42 is 42. Therefore, we write 42 to the left of the top row. You can see it here in the color green:

42  42x 42
48x  48x² 48x
x 1
Next, let’s look at the bottom row. We have the terms 48x² and 48x. The greatest common factor of 48x² and 48x is 48x. Therefore, we write 48x to the left of the bottom row. You can see it here in the color blue:

42  42x 42
48x  48x² 48x
x 1
To find the values below the table, we first divide 48x² by 48x (labeled in blue). This gives us x.

48x² ÷ 48x = x

You can see this value colored in orange below:

42  42x 42
48x  48x² 48x
x 1

Next, we divide 48x by 48x (labeled in blue). This gives us 1.

48x ÷ 48x = 1

You can see this value colored in purple below:

42  42x 42
48x  48x² 48x
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 -48x² - 90x - 42. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(48x + 42)(x + 1)

In our original quadratic equation, -48x² - 90x - 42, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(48x + 42)(x + 1)

That’s it! Now you know how to factor the equation -48x² - 90x - 42.


Factoring Quadratics
Go here if you want to learn how to factor another quadratic function.

Factor -48x² - 90x - 33
Here is the next quadratic function on our list that we have factored for you.


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