Factor 19x² - 90x + 56

Here we will show you how to factor the quadratic function 19x² - 90x + 56 using the box method. In other words, we will show you how to factor 19x squared minus 90x plus 56 (19x^2 - 90x + 56) 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 19x² - 90x + 56, like this:

a = 19
b = -90
c = 56

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

-4	-76x	56
x	19x²	-14x
	19x	-14

We put 19x² (a) in the bottom left square and 56 (c) in the top right square, like this:

-4	-76x	56
x	19x²	-14x
	19x	-14

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 19 times 56 (a × c), and add together to equal -90 (b).

More specifically, 19 times 56 is 1064. Therefore, we need to find the two numbers that multiply to equal 1064, and add to equal -90.

? × ? = 1064
? + ? = -90

After looking at this problem, we can see that the two numbers that multiply together to equal 1064, and add together to equal -90, are -76 and -14, as illustrated here:

-76 × -14 = 1064
-76 + -14 = -90

Now, we can fill in the last two squares in our box with -76x and -14x. Place -76x in the upper left square, and place -14x in the lower right square.

-4	-76x	56
x	19x²	-14x
	19x	-14

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 -76x and 56. The greatest common factor of -76x and 56 is -4. Therefore, we write -4 to the left of the top row. You can see it here in the color green:

-4	-76x	56
x	19x²	-14x
	19x	-14

Next, let’s look at the bottom row. We have the terms 19x² and -14x. The greatest common factor of 19x² and -14x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-4	-76x	56
x	19x²	-14x
	19x	-14

To find the values below the table, we first divide 19x² by x (labeled in blue). This gives us 19x.

19x² ÷ x = 19x

You can see this value colored in orange below:

-4	-76x	56
x	19x²	-14x
	19x	-14

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

-14x ÷ x = -14

You can see this value colored in purple below:

-4	-76x	56
x	19x²	-14x
	19x	-14

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 19x² - 90x + 56. 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:

(x - 4)(19x - 14)

That’s it! Now you know how to factor the equation 19x² - 90x + 56.

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