Factor 34x² - 89x + 42

Here we will show you how to factor the quadratic function 34x² - 89x + 42 using the box method. In other words, we will show you how to factor 34x squared minus 89x plus 42 (34x^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 34x² - 89x + 42, like this:

a = 34
b = -89
c = 42

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

-2	-68x	42
x	34x²	-21x
	34x	-21

We put 34x² (a) in the bottom left square and 42 (c) in the top right square, like this:

-2	-68x	42
x	34x²	-21x
	34x	-21

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

More specifically, 34 times 42 is 1428. Therefore, we need to find the two numbers that multiply to equal 1428, and add to equal -89.

? × ? = 1428
? + ? = -89

After looking at this problem, we can see that the two numbers that multiply together to equal 1428, and add together to equal -89, are -68 and -21, as illustrated here:

-68 × -21 = 1428
-68 + -21 = -89

Now, we can fill in the last two squares in our box with -68x and -21x. Place -68x in the upper left square, and place -21x in the lower right square.

-2	-68x	42
x	34x²	-21x
	34x	-21

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 -68x and 42. The greatest common factor of -68x and 42 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-68x	42
x	34x²	-21x
	34x	-21

Next, let’s look at the bottom row. We have the terms 34x² and -21x. The greatest common factor of 34x² and -21x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-2	-68x	42
x	34x²	-21x
	34x	-21

To find the values below the table, we first divide 34x² by x (labeled in blue). This gives us 34x.

34x² ÷ x = 34x

You can see this value colored in orange below:

-2	-68x	42
x	34x²	-21x
	34x	-21

Next, we divide -21x by x (labeled in blue). This gives us -21.

-21x ÷ x = -21

You can see this value colored in purple below:

-2	-68x	42
x	34x²	-21x
	34x	-21

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 34x² - 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 - 2)(34x - 21)

That’s it! Now you know how to factor the equation 34x² - 89x + 42.

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