Factor 35x² + 64x + 29

Here we will show you how to factor the quadratic function 35x² + 64x + 29 using the box method. In other words, we will show you how to factor 35x squared plus 64x plus 29 (35x^2 + 64x + 29) 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 + 64x + 29, like this:

a = 35
b = 64
c = 29

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

29	29x	29
35x	35x²	35x
	x	1

We put 35x² (a) in the bottom left square and 29 (c) in the top right square, like this:

29	29x	29
35x	35x²	35x
	x	1

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 29 (a × c), and add together to equal 64 (b).

More specifically, 35 times 29 is 1015. Therefore, we need to find the two numbers that multiply to equal 1015, and add to equal 64.

? × ? = 1015
? + ? = 64

After looking at this problem, we can see that the two numbers that multiply together to equal 1015, and add together to equal 64, are 29 and 35, as illustrated here:

29 × 35 = 1015
29 + 35 = 64

Now, we can fill in the last two squares in our box with 29x and 35x. Place 29x in the upper left square, and place 35x in the lower right square.

29	29x	29
35x	35x²	35x
	x	1

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 29x and 29. The greatest common factor of 29x and 29 is 29. Therefore, we write 29 to the left of the top row. You can see it here in the color green:

29	29x	29
35x	35x²	35x
	x	1

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

29	29x	29
35x	35x²	35x
	x	1

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

35x² ÷ 35x = x

You can see this value colored in orange below:

29	29x	29
35x	35x²	35x
	x	1

Next, we divide 35x by 35x (labeled in blue). This gives us 1.

35x ÷ 35x = 1

You can see this value colored in purple below:

29	29x	29
35x	35x²	35x
	x	1

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 + 64x + 29. 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:

(35x + 29)(x + 1)

That’s it! Now you know how to factor the equation 35x² + 64x + 29.

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