Factor 65x² - 98x

Factor 65x² - 98x - 99

Here we will show you how to factor the quadratic function 65x² - 98x - 99 using the box method. In other words, we will show you how to factor 65x squared minus 98x minus 99 (65x^2 - 98x - 99) 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 65x² - 98x - 99, like this:

a = 65
b = -98
c = -99

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

-11	-143x	-99
5x	65x²	45x
	13x	9

We put 65x² (a) in the bottom left square and -99 (c) in the top right square, like this:

-11	-143x	-99
5x	65x²	45x
	13x	9

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 65 times -99 (a × c), and add together to equal -98 (b).

More specifically, 65 times -99 is -6435. Therefore, we need to find the two numbers that multiply to equal -6435, and add to equal -98.

? × ? = -6435
? + ? = -98

After looking at this problem, we can see that the two numbers that multiply together to equal -6435, and add together to equal -98, are -143 and 45, as illustrated here:

-143 × 45 = -6435
-143 + 45 = -98

Now, we can fill in the last two squares in our box with -143x and 45x. Place -143x in the upper left square, and place 45x in the lower right square.

-11	-143x	-99
5x	65x²	45x
	13x	9

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 -143x and -99. The greatest common factor of -143x and -99 is -11. Therefore, we write -11 to the left of the top row. You can see it here in the color green:

-11	-143x	-99
5x	65x²	45x
	13x	9

Next, let’s look at the bottom row. We have the terms 65x² and 45x. The greatest common factor of 65x² and 45x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

-11	-143x	-99
5x	65x²	45x
	13x	9

To find the values below the table, we first divide 65x² by 5x (labeled in blue). This gives us 13x.

65x² ÷ 5x = 13x

You can see this value colored in orange below:

-11	-143x	-99
5x	65x²	45x
	13x	9

Next, we divide 45x by 5x (labeled in blue). This gives us 9.

45x ÷ 5x = 9

You can see this value colored in purple below:

-11	-143x	-99
5x	65x²	45x
	13x	9

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 65x² - 98x - 99. 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 - 11)(13x + 9)

That’s it! Now you know how to factor the equation 65x² - 98x - 99.

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