Factor 47x² - 89x + 42

Here we will show you how to factor the quadratic function 47x² - 89x + 42 using the box method. In other words, we will show you how to factor 47x squared minus 89x plus 42 (47x^2 - 89x + 42) 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 47x² - 89x + 42, like this:

a = 47
b = -89
c = 42

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

-1	-47x	42
x	47x²	-42x
	47x	-42

We put 47x² (a) in the bottom left square and 42 (c) in the top right square, like this:

-1	-47x	42
x	47x²	-42x
	47x	-42

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 47 times 42 (a × c), and add together to equal -89 (b).

More specifically, 47 times 42 is 1974. Therefore, we need to find the two numbers that multiply to equal 1974, and add to equal -89.

? × ? = 1974
? + ? = -89

After looking at this problem, we can see that the two numbers that multiply together to equal 1974, and add together to equal -89, are -47 and -42, as illustrated here:

-47 × -42 = 1974
-47 + -42 = -89

Now, we can fill in the last two squares in our box with -47x and -42x. Place -47x in the upper left square, and place -42x in the lower right square.

-1	-47x	42
x	47x²	-42x
	47x	-42

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 -47x and 42. The greatest common factor of -47x and 42 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-47x	42
x	47x²	-42x
	47x	-42

Next, let’s look at the bottom row. We have the terms 47x² and -42x. The greatest common factor of 47x² and -42x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-47x	42
x	47x²	-42x
	47x	-42

To find the values below the table, we first divide 47x² by x (labeled in blue). This gives us 47x.

47x² ÷ x = 47x

You can see this value colored in orange below:

-1	-47x	42
x	47x²	-42x
	47x	-42

Next, we divide -42x by x (labeled in blue). This gives us -42.

-42x ÷ x = -42

You can see this value colored in purple below:

-1	-47x	42
x	47x²	-42x
	47x	-42

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 47x² - 89x + 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 the answer:

(x - 1)(47x - 42)

That’s it! Now you know how to factor the equation 47x² - 89x + 42.

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