Factor 40x² + 38x

Factor 40x² + 38x - 15

Here we will show you how to factor the quadratic function 40x² + 38x - 15 using the box method. In other words, we will show you how to factor 40x squared plus 38x minus 15 (40x^2 + 38x - 15) 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 40x² + 38x - 15, like this:

a = 40
b = 38
c = -15

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

-3	-12x	-15
10x	40x²	50x
	4x	5

We put 40x² (a) in the bottom left square and -15 (c) in the top right square, like this:

-3	-12x	-15
10x	40x²	50x
	4x	5

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 40 times -15 (a × c), and add together to equal 38 (b).

More specifically, 40 times -15 is -600. Therefore, we need to find the two numbers that multiply to equal -600, and add to equal 38.

? × ? = -600
? + ? = 38

After looking at this problem, we can see that the two numbers that multiply together to equal -600, and add together to equal 38, are -12 and 50, as illustrated here:

-12 × 50 = -600
-12 + 50 = 38

Now, we can fill in the last two squares in our box with -12x and 50x. Place -12x in the upper left square, and place 50x in the lower right square.

-3	-12x	-15
10x	40x²	50x
	4x	5

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 -12x and -15. The greatest common factor of -12x and -15 is -3. Therefore, we write -3 to the left of the top row. You can see it here in the color green:

-3	-12x	-15
10x	40x²	50x
	4x	5

Next, let’s look at the bottom row. We have the terms 40x² and 50x. The greatest common factor of 40x² and 50x is 10x. Therefore, we write 10x to the left of the bottom row. You can see it here in the color blue:

-3	-12x	-15
10x	40x²	50x
	4x	5

To find the values below the table, we first divide 40x² by 10x (labeled in blue). This gives us 4x.

40x² ÷ 10x = 4x

You can see this value colored in orange below:

-3	-12x	-15
10x	40x²	50x
	4x	5

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

50x ÷ 10x = 5

You can see this value colored in purple below:

-3	-12x	-15
10x	40x²	50x
	4x	5

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 40x² + 38x - 15. 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:

(10x - 3)(4x + 5)

That’s it! Now you know how to factor the equation 40x² + 38x - 15.

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