Factor -45x² - 43x

Factor -45x² - 43x - 10

Here we will show you how to factor the quadratic function -45x² - 43x - 10 using the box method. In other words, we will show you how to factor negative 45x squared minus 43x minus 10 (-45x^2 - 43x - 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 + 43x + 10. Now we can label the different parts of our equation, like this:

a = 45
b = 43
c = 10

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

2	18x	10
5x	45x²	25x
	9x	5

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

2	18x	10
5x	45x²	25x
	9x	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 45 times 10 (a × c), and add together to equal 43 (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 43.

? × ? = 450
? + ? = 43

After looking at this problem, we can see that the two numbers that multiply together to equal 450, and add together to equal 43, are 18 and 25, as illustrated here:

18 × 25 = 450
18 + 25 = 43

Now, we can fill in the last two squares in our box with 18x and 25x. Place 18x in the upper left square, and place 25x in the lower right square.

2	18x	10
5x	45x²	25x
	9x	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 18x and 10. The greatest common factor of 18x and 10 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	18x	10
5x	45x²	25x
	9x	5

Next, let’s look at the bottom row. We have the terms 45x² and 25x. The greatest common factor of 45x² and 25x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

2	18x	10
5x	45x²	25x
	9x	5

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:

2	18x	10
5x	45x²	25x
	9x	5

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

25x ÷ 5x = 5

You can see this value colored in purple below:

2	18x	10
5x	45x²	25x
	9x	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 -45x² - 43x - 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:

(5x + 2)(9x + 5)

In our original quadratic equation, -45x² - 43x - 10, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(5x + 2)(9x + 5)

That’s it! Now you know how to factor the equation -45x² - 43x - 10.

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