Factor 3x² - 34x + 40

Here we will show you how to factor the quadratic function 3x² - 34x + 40 using the box method. In other words, we will show you how to factor 3x squared minus 34x plus 40 (3x^2 - 34x + 40) 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 3x² - 34x + 40, like this:

a = 3
b = -34
c = 40

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

-10	-30x	40
x	3x²	-4x
	3x	-4

We put 3x² (a) in the bottom left square and 40 (c) in the top right square, like this:

-10	-30x	40
x	3x²	-4x
	3x	-4

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 3 times 40 (a × c), and add together to equal -34 (b).

More specifically, 3 times 40 is 120. Therefore, we need to find the two numbers that multiply to equal 120, and add to equal -34.

? × ? = 120
? + ? = -34

After looking at this problem, we can see that the two numbers that multiply together to equal 120, and add together to equal -34, are -30 and -4, as illustrated here:

-30 × -4 = 120
-30 + -4 = -34

Now, we can fill in the last two squares in our box with -30x and -4x. Place -30x in the upper left square, and place -4x in the lower right square.

-10	-30x	40
x	3x²	-4x
	3x	-4

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 -30x and 40. The greatest common factor of -30x and 40 is -10. Therefore, we write -10 to the left of the top row. You can see it here in the color green:

-10	-30x	40
x	3x²	-4x
	3x	-4

Next, let’s look at the bottom row. We have the terms 3x² and -4x. The greatest common factor of 3x² and -4x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-10	-30x	40
x	3x²	-4x
	3x	-4

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

3x² ÷ x = 3x

You can see this value colored in orange below:

-10	-30x	40
x	3x²	-4x
	3x	-4

Next, we divide -4x by x (labeled in blue). This gives us -4.

-4x ÷ x = -4

You can see this value colored in purple below:

-10	-30x	40
x	3x²	-4x
	3x	-4

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 3x² - 34x + 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 the answer:

(x - 10)(3x - 4)

That’s it! Now you know how to factor the equation 3x² - 34x + 40.

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