Factor 10x² + 85x + 75

Here we will show you how to factor the quadratic function 10x² + 85x + 75 using the box method. In other words, we will show you how to factor 10x squared plus 85x plus 75 (10x^2 + 85x + 75) 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 10x² + 85x + 75, like this:

a = 10
b = 85
c = 75

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

5	10x	75
5x	10x²	75x
	2x	15

We put 10x² (a) in the bottom left square and 75 (c) in the top right square, like this:

5	10x	75
5x	10x²	75x
	2x	15

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 10 times 75 (a × c), and add together to equal 85 (b).

More specifically, 10 times 75 is 750. Therefore, we need to find the two numbers that multiply to equal 750, and add to equal 85.

? × ? = 750
? + ? = 85

After looking at this problem, we can see that the two numbers that multiply together to equal 750, and add together to equal 85, are 10 and 75, as illustrated here:

10 × 75 = 750
10 + 75 = 85

Now, we can fill in the last two squares in our box with 10x and 75x. Place 10x in the upper left square, and place 75x in the lower right square.

5	10x	75
5x	10x²	75x
	2x	15

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 10x and 75. The greatest common factor of 10x and 75 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	10x	75
5x	10x²	75x
	2x	15

Next, let’s look at the bottom row. We have the terms 10x² and 75x. The greatest common factor of 10x² and 75x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

5	10x	75
5x	10x²	75x
	2x	15

To find the values below the table, we first divide 10x² by 5x (labeled in blue). This gives us 2x.

10x² ÷ 5x = 2x

You can see this value colored in orange below:

5	10x	75
5x	10x²	75x
	2x	15

Next, we divide 75x by 5x (labeled in blue). This gives us 15.

75x ÷ 5x = 15

You can see this value colored in purple below:

5	10x	75
5x	10x²	75x
	2x	15

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 10x² + 85x + 75. 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:

(5x + 5)(2x + 15)

That’s it! Now you know how to factor the equation 10x² + 85x + 75.

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