Factor 45x² + 98x

Factor 45x² + 98x - 75

Here we will show you how to factor the quadratic function 45x² + 98x - 75 using the box method. In other words, we will show you how to factor 45x squared plus 98x minus 75 (45x^2 + 98x - 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 45x² + 98x - 75, like this:

a = 45
b = 98
c = -75

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

-3	-27x	-75
5x	45x²	125x
	9x	25

We put 45x² (a) in the bottom left square and -75 (c) in the top right square, like this:

-3	-27x	-75
5x	45x²	125x
	9x	25

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 45 times -75 (a × c), and add together to equal 98 (b).

More specifically, 45 times -75 is -3375. Therefore, we need to find the two numbers that multiply to equal -3375, and add to equal 98.

? × ? = -3375
? + ? = 98

After looking at this problem, we can see that the two numbers that multiply together to equal -3375, and add together to equal 98, are -27 and 125, as illustrated here:

-27 × 125 = -3375
-27 + 125 = 98

Now, we can fill in the last two squares in our box with -27x and 125x. Place -27x in the upper left square, and place 125x in the lower right square.

-3	-27x	-75
5x	45x²	125x
	9x	25

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 -27x and -75. The greatest common factor of -27x and -75 is -3. Therefore, we write -3 to the left of the top row. You can see it here in the color green:

-3	-27x	-75
5x	45x²	125x
	9x	25

Next, let’s look at the bottom row. We have the terms 45x² and 125x. The greatest common factor of 45x² and 125x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

-3	-27x	-75
5x	45x²	125x
	9x	25

To find the values below the table, we first divide 45x² by 5x (labeled in blue). This gives us 9x.

45x² ÷ 5x = 9x

You can see this value colored in orange below:

-3	-27x	-75
5x	45x²	125x
	9x	25

Next, we divide 125x by 5x (labeled in blue). This gives us 25.

125x ÷ 5x = 25

You can see this value colored in purple below:

-3	-27x	-75
5x	45x²	125x
	9x	25

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 45x² + 98x - 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:

(5x - 3)(9x + 25)

That’s it! Now you know how to factor the equation 45x² + 98x - 75.

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