Factor 7x² - 41x + 30

Here we will show you how to factor the quadratic function 7x² - 41x + 30 using the box method. In other words, we will show you how to factor 7x squared minus 41x plus 30 (7x^2 - 41x + 30) 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 7x² - 41x + 30, like this:

a = 7
b = -41
c = 30

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

-5	-35x	30
x	7x²	-6x
	7x	-6

We put 7x² (a) in the bottom left square and 30 (c) in the top right square, like this:

-5	-35x	30
x	7x²	-6x
	7x	-6

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 7 times 30 (a × c), and add together to equal -41 (b).

More specifically, 7 times 30 is 210. Therefore, we need to find the two numbers that multiply to equal 210, and add to equal -41.

? × ? = 210
? + ? = -41

After looking at this problem, we can see that the two numbers that multiply together to equal 210, and add together to equal -41, are -35 and -6, as illustrated here:

-35 × -6 = 210
-35 + -6 = -41

Now, we can fill in the last two squares in our box with -35x and -6x. Place -35x in the upper left square, and place -6x in the lower right square.

-5	-35x	30
x	7x²	-6x
	7x	-6

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 -35x and 30. The greatest common factor of -35x and 30 is -5. Therefore, we write -5 to the left of the top row. You can see it here in the color green:

-5	-35x	30
x	7x²	-6x
	7x	-6

Next, let’s look at the bottom row. We have the terms 7x² and -6x. The greatest common factor of 7x² and -6x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-5	-35x	30
x	7x²	-6x
	7x	-6

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

7x² ÷ x = 7x

You can see this value colored in orange below:

-5	-35x	30
x	7x²	-6x
	7x	-6

Next, we divide -6x by x (labeled in blue). This gives us -6.

-6x ÷ x = -6

You can see this value colored in purple below:

-5	-35x	30
x	7x²	-6x
	7x	-6

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 7x² - 41x + 30. 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 - 5)(7x - 6)

That’s it! Now you know how to factor the equation 7x² - 41x + 30.

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