Factor 40x² - 94x + 21

Here we will show you how to factor the quadratic function 40x² - 94x + 21 using the box method. In other words, we will show you how to factor 40x squared minus 94x plus 21 (40x^2 - 94x + 21) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 40x² - 94x + 21, like this:

a = 40
b = -94
c = 21

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

-21	-84x	21
10x	40x²	-10x
	4x	-1

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

-21	-84x	21
10x	40x²	-10x
	4x	-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 40 times 21 (a × c), and add together to equal -94 (b).

More specifically, 40 times 21 is 840. Therefore, we need to find the two numbers that multiply to equal 840, and add to equal -94.

? × ? = 840
? + ? = -94

After looking at this problem, we can see that the two numbers that multiply together to equal 840, and add together to equal -94, are -84 and -10, as illustrated here:

-84 × -10 = 840
-84 + -10 = -94

Now, we can fill in the last two squares in our box with -84x and -10x. Place -84x in the upper left square, and place -10x in the lower right square.

-21	-84x	21
10x	40x²	-10x
	4x	-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 -84x and 21. The greatest common factor of -84x and 21 is -21. Therefore, we write -21 to the left of the top row. You can see it here in the color green:

-21	-84x	21
10x	40x²	-10x
	4x	-1

Next, let’s look at the bottom row. We have the terms 40x² and -10x. The greatest common factor of 40x² and -10x is 10x. Therefore, we write 10x to the left of the bottom row. You can see it here in the color blue:

-21	-84x	21
10x	40x²	-10x
	4x	-1

To find the values below the table, we first divide 40x² by 10x (labeled in blue). This gives us 4x.

40x² ÷ 10x = 4x

You can see this value colored in orange below:

-21	-84x	21
10x	40x²	-10x
	4x	-1

Next, we divide -10x by 10x (labeled in blue). This gives us -1.

-10x ÷ 10x = -1

You can see this value colored in purple below:

-21	-84x	21
10x	40x²	-10x
	4x	-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 40x² - 94x + 21. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(10x - 21)(4x - 1)

That’s it! Now you know how to factor the equation 40x² - 94x + 21.

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