Factor 14x² + 72x + 90

Here we will show you how to factor the quadratic function 14x² + 72x + 90 using the box method. In other words, we will show you how to factor 14x squared plus 72x plus 90 (14x^2 + 72x + 90) 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 14x² + 72x + 90, like this:

a = 14
b = 72
c = 90

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

30	30x	90
14x	14x²	42x
	x	3

We put 14x² (a) in the bottom left square and 90 (c) in the top right square, like this:

30	30x	90
14x	14x²	42x
	x	3

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 14 times 90 (a × c), and add together to equal 72 (b).

More specifically, 14 times 90 is 1260. Therefore, we need to find the two numbers that multiply to equal 1260, and add to equal 72.

? × ? = 1260
? + ? = 72

After looking at this problem, we can see that the two numbers that multiply together to equal 1260, and add together to equal 72, are 30 and 42, as illustrated here:

30 × 42 = 1260
30 + 42 = 72

Now, we can fill in the last two squares in our box with 30x and 42x. Place 30x in the upper left square, and place 42x in the lower right square.

30	30x	90
14x	14x²	42x
	x	3

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 90. The greatest common factor of 30x and 90 is 30. Therefore, we write 30 to the left of the top row. You can see it here in the color green:

30	30x	90
14x	14x²	42x
	x	3

Next, let’s look at the bottom row. We have the terms 14x² and 42x. The greatest common factor of 14x² and 42x is 14x. Therefore, we write 14x to the left of the bottom row. You can see it here in the color blue:

30	30x	90
14x	14x²	42x
	x	3

To find the values below the table, we first divide 14x² by 14x (labeled in blue). This gives us x.

14x² ÷ 14x = x

You can see this value colored in orange below:

30	30x	90
14x	14x²	42x
	x	3

Next, we divide 42x by 14x (labeled in blue). This gives us 3.

42x ÷ 14x = 3

You can see this value colored in purple below:

30	30x	90
14x	14x²	42x
	x	3

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 14x² + 72x + 90. 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:

(14x + 30)(x + 3)

That’s it! Now you know how to factor the equation 14x² + 72x + 90.

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