Factor 87x² + 100x

Factor 87x² + 100x - 75

Here we will show you how to factor the quadratic function 87x² + 100x - 75 using the box method. In other words, we will show you how to factor 87x squared plus 100x minus 75 (87x^2 + 100x - 75) 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 87x² + 100x - 75, like this:

a = 87
b = 100
c = -75

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

-15	-45x	-75
29x	87x²	145x
	3x	5

We put 87x² (a) in the bottom left square and -75 (c) in the top right square, like this:

-15	-45x	-75
29x	87x²	145x
	3x	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 87 times -75 (a × c), and add together to equal 100 (b).

More specifically, 87 times -75 is -6525. Therefore, we need to find the two numbers that multiply to equal -6525, and add to equal 100.

? × ? = -6525
? + ? = 100

After looking at this problem, we can see that the two numbers that multiply together to equal -6525, and add together to equal 100, are -45 and 145, as illustrated here:

-45 × 145 = -6525
-45 + 145 = 100

Now, we can fill in the last two squares in our box with -45x and 145x. Place -45x in the upper left square, and place 145x in the lower right square.

-15	-45x	-75
29x	87x²	145x
	3x	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 -45x and -75. The greatest common factor of -45x and -75 is -15. Therefore, we write -15 to the left of the top row. You can see it here in the color green:

-15	-45x	-75
29x	87x²	145x
	3x	5

Next, let’s look at the bottom row. We have the terms 87x² and 145x. The greatest common factor of 87x² and 145x is 29x. Therefore, we write 29x to the left of the bottom row. You can see it here in the color blue:

-15	-45x	-75
29x	87x²	145x
	3x	5

To find the values below the table, we first divide 87x² by 29x (labeled in blue). This gives us 3x.

87x² ÷ 29x = 3x

You can see this value colored in orange below:

-15	-45x	-75
29x	87x²	145x
	3x	5

Next, we divide 145x by 29x (labeled in blue). This gives us 5.

145x ÷ 29x = 5

You can see this value colored in purple below:

-15	-45x	-75
29x	87x²	145x
	3x	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 87x² + 100x - 75. 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:

(29x - 15)(3x + 5)

That’s it! Now you know how to factor the equation 87x² + 100x - 75.

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