Factor -45x² - 45x

Factor -45x² - 45x - 10

Here we will show you how to factor the quadratic function -45x² - 45x - 10 using the box method. In other words, we will show you how to factor negative 45x squared minus 45x minus 10 (-45x^2 - 45x - 10) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 45x² + 45x + 10. Now we can label the different parts of our equation, like this:

a = 45
b = 45
c = 10

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

5	15x	10
15x	45x²	30x
	3x	2

We put 45x² (a) in the bottom left square and 10 (c) in the top right square, like this:

5	15x	10
15x	45x²	30x
	3x	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 45 times 10 (a × c), and add together to equal 45 (b).

More specifically, 45 times 10 is 450. Therefore, we need to find the two numbers that multiply to equal 450, and add to equal 45.

? × ? = 450
? + ? = 45

After looking at this problem, we can see that the two numbers that multiply together to equal 450, and add together to equal 45, are 15 and 30, as illustrated here:

15 × 30 = 450
15 + 30 = 45

Now, we can fill in the last two squares in our box with 15x and 30x. Place 15x in the upper left square, and place 30x in the lower right square.

5	15x	10
15x	45x²	30x
	3x	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 15x and 10. The greatest common factor of 15x and 10 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	15x	10
15x	45x²	30x
	3x	2

Next, let’s look at the bottom row. We have the terms 45x² and 30x. The greatest common factor of 45x² and 30x is 15x. Therefore, we write 15x to the left of the bottom row. You can see it here in the color blue:

5	15x	10
15x	45x²	30x
	3x	2

To find the values below the table, we first divide 45x² by 15x (labeled in blue). This gives us 3x.

45x² ÷ 15x = 3x

You can see this value colored in orange below:

5	15x	10
15x	45x²	30x
	3x	2

Next, we divide 30x by 15x (labeled in blue). This gives us 2.

30x ÷ 15x = 2

You can see this value colored in purple below:

5	15x	10
15x	45x²	30x
	3x	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 -45x² - 45x - 10. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(15x + 5)(3x + 2)

In our original quadratic equation, -45x² - 45x - 10, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(15x + 5)(3x + 2)

That’s it! Now you know how to factor the equation -45x² - 45x - 10.

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