Factor 35x² + 99x + 70

Here we will show you how to factor the quadratic function 35x² + 99x + 70 using the box method. In other words, we will show you how to factor 35x squared plus 99x plus 70 (35x^2 + 99x + 70) 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 35x² + 99x + 70, like this:

a = 35
b = 99
c = 70

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

7	49x	70
5x	35x²	50x
	7x	10

We put 35x² (a) in the bottom left square and 70 (c) in the top right square, like this:

7	49x	70
5x	35x²	50x
	7x	10

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 35 times 70 (a × c), and add together to equal 99 (b).

More specifically, 35 times 70 is 2450. Therefore, we need to find the two numbers that multiply to equal 2450, and add to equal 99.

? × ? = 2450
? + ? = 99

After looking at this problem, we can see that the two numbers that multiply together to equal 2450, and add together to equal 99, are 49 and 50, as illustrated here:

49 × 50 = 2450
49 + 50 = 99

Now, we can fill in the last two squares in our box with 49x and 50x. Place 49x in the upper left square, and place 50x in the lower right square.

7	49x	70
5x	35x²	50x
	7x	10

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 49x and 70. The greatest common factor of 49x and 70 is 7. Therefore, we write 7 to the left of the top row. You can see it here in the color green:

7	49x	70
5x	35x²	50x
	7x	10

Next, let’s look at the bottom row. We have the terms 35x² and 50x. The greatest common factor of 35x² and 50x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

7	49x	70
5x	35x²	50x
	7x	10

To find the values below the table, we first divide 35x² by 5x (labeled in blue). This gives us 7x.

35x² ÷ 5x = 7x

You can see this value colored in orange below:

7	49x	70
5x	35x²	50x
	7x	10

Next, we divide 50x by 5x (labeled in blue). This gives us 10.

50x ÷ 5x = 10

You can see this value colored in purple below:

7	49x	70
5x	35x²	50x
	7x	10

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 35x² + 99x + 70. 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 + 7)(7x + 10)

That’s it! Now you know how to factor the equation 35x² + 99x + 70.

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