Factor 13x² - 19x

Factor 13x² - 19x - 90

Here we will show you how to factor the quadratic function 13x² - 19x - 90 using the box method. In other words, we will show you how to factor 13x squared minus 19x minus 90 (13x^2 - 19x - 90) 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 13x² - 19x - 90, like this:

a = 13
b = -19
c = -90

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

-45	-45x	-90
13x	13x²	26x
	x	2

We put 13x² (a) in the bottom left square and -90 (c) in the top right square, like this:

-45	-45x	-90
13x	13x²	26x
	x	2

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 13 times -90 (a × c), and add together to equal -19 (b).

More specifically, 13 times -90 is -1170. Therefore, we need to find the two numbers that multiply to equal -1170, and add to equal -19.

? × ? = -1170
? + ? = -19

After looking at this problem, we can see that the two numbers that multiply together to equal -1170, and add together to equal -19, are -45 and 26, as illustrated here:

-45 × 26 = -1170
-45 + 26 = -19

Now, we can fill in the last two squares in our box with -45x and 26x. Place -45x in the upper left square, and place 26x in the lower right square.

-45	-45x	-90
13x	13x²	26x
	x	2

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 -45x and -90. The greatest common factor of -45x and -90 is -45. Therefore, we write -45 to the left of the top row. You can see it here in the color green:

-45	-45x	-90
13x	13x²	26x
	x	2

Next, let’s look at the bottom row. We have the terms 13x² and 26x. The greatest common factor of 13x² and 26x is 13x. Therefore, we write 13x to the left of the bottom row. You can see it here in the color blue:

-45	-45x	-90
13x	13x²	26x
	x	2

To find the values below the table, we first divide 13x² by 13x (labeled in blue). This gives us x.

13x² ÷ 13x = x

You can see this value colored in orange below:

-45	-45x	-90
13x	13x²	26x
	x	2

Next, we divide 26x by 13x (labeled in blue). This gives us 2.

26x ÷ 13x = 2

You can see this value colored in purple below:

-45	-45x	-90
13x	13x²	26x
	x	2

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 13x² - 19x - 90. 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:

(13x - 45)(x + 2)

That’s it! Now you know how to factor the equation 13x² - 19x - 90.

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