Factor -21x² - 58x

Factor -21x² - 58x - 40

Here we will show you how to factor the quadratic function -21x² - 58x - 40 using the box method. In other words, we will show you how to factor negative 21x squared minus 58x minus 40 (-21x^2 - 58x - 40) 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 21x² + 58x + 40. Now we can label the different parts of our equation, like this:

a = 21
b = 58
c = 40

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

4	28x	40
3x	21x²	30x
	7x	10

We put 21x² (a) in the bottom left square and 40 (c) in the top right square, like this:

4	28x	40
3x	21x²	30x
	7x	10

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 21 times 40 (a × c), and add together to equal 58 (b).

More specifically, 21 times 40 is 840. Therefore, we need to find the two numbers that multiply to equal 840, and add to equal 58.

? × ? = 840
? + ? = 58

After looking at this problem, we can see that the two numbers that multiply together to equal 840, and add together to equal 58, are 28 and 30, as illustrated here:

28 × 30 = 840
28 + 30 = 58

Now, we can fill in the last two squares in our box with 28x and 30x. Place 28x in the upper left square, and place 30x in the lower right square.

4	28x	40
3x	21x²	30x
	7x	10

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 28x and 40. The greatest common factor of 28x and 40 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	28x	40
3x	21x²	30x
	7x	10

Next, let’s look at the bottom row. We have the terms 21x² and 30x. The greatest common factor of 21x² and 30x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

4	28x	40
3x	21x²	30x
	7x	10

To find the values below the table, we first divide 21x² by 3x (labeled in blue). This gives us 7x.

21x² ÷ 3x = 7x

You can see this value colored in orange below:

4	28x	40
3x	21x²	30x
	7x	10

Next, we divide 30x by 3x (labeled in blue). This gives us 10.

30x ÷ 3x = 10

You can see this value colored in purple below:

4	28x	40
3x	21x²	30x
	7x	10

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 -21x² - 58x - 40. 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 + 4)(7x + 10)

In our original quadratic equation, -21x² - 58x - 40, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x + 4)(7x + 10)

That’s it! Now you know how to factor the equation -21x² - 58x - 40.

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