Factor 45x² + 93x + 42

Here we will show you how to factor the quadratic function 45x² + 93x + 42 using the box method. In other words, we will show you how to factor 45x squared plus 93x plus 42 (45x^2 + 93x + 42) 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 + 93x + 42, like this:

a = 45
b = 93
c = 42

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

6	30x	42
9x	45x²	63x
	5x	7

We put 45x² (a) in the bottom left square and 42 (c) in the top right square, like this:

6	30x	42
9x	45x²	63x
	5x	7

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 42 (a × c), and add together to equal 93 (b).

More specifically, 45 times 42 is 1890. Therefore, we need to find the two numbers that multiply to equal 1890, and add to equal 93.

? × ? = 1890
? + ? = 93

After looking at this problem, we can see that the two numbers that multiply together to equal 1890, and add together to equal 93, are 30 and 63, as illustrated here:

30 × 63 = 1890
30 + 63 = 93

Now, we can fill in the last two squares in our box with 30x and 63x. Place 30x in the upper left square, and place 63x in the lower right square.

6	30x	42
9x	45x²	63x
	5x	7

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 30x and 42. The greatest common factor of 30x and 42 is 6. Therefore, we write 6 to the left of the top row. You can see it here in the color green:

6	30x	42
9x	45x²	63x
	5x	7

Next, let’s look at the bottom row. We have the terms 45x² and 63x. The greatest common factor of 45x² and 63x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

6	30x	42
9x	45x²	63x
	5x	7

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

45x² ÷ 9x = 5x

You can see this value colored in orange below:

6	30x	42
9x	45x²	63x
	5x	7

Next, we divide 63x by 9x (labeled in blue). This gives us 7.

63x ÷ 9x = 7

You can see this value colored in purple below:

6	30x	42
9x	45x²	63x
	5x	7

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 + 93x + 42. 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:

(9x + 6)(5x + 7)

That’s it! Now you know how to factor the equation 45x² + 93x + 42.

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