Factor 98x² - 95x + 23

Here we will show you how to factor the quadratic function 98x² - 95x + 23 using the box method. In other words, we will show you how to factor 98x squared minus 95x plus 23 (98x^2 - 95x + 23) 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 98x² - 95x + 23, like this:

a = 98
b = -95
c = 23

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

-1	-49x	23
2x	98x²	-46x
	49x	-23

We put 98x² (a) in the bottom left square and 23 (c) in the top right square, like this:

-1	-49x	23
2x	98x²	-46x
	49x	-23

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 98 times 23 (a × c), and add together to equal -95 (b).

More specifically, 98 times 23 is 2254. Therefore, we need to find the two numbers that multiply to equal 2254, and add to equal -95.

? × ? = 2254
? + ? = -95

After looking at this problem, we can see that the two numbers that multiply together to equal 2254, and add together to equal -95, are -49 and -46, as illustrated here:

-49 × -46 = 2254
-49 + -46 = -95

Now, we can fill in the last two squares in our box with -49x and -46x. Place -49x in the upper left square, and place -46x in the lower right square.

-1	-49x	23
2x	98x²	-46x
	49x	-23

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 -49x and 23. The greatest common factor of -49x and 23 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-49x	23
2x	98x²	-46x
	49x	-23

Next, let’s look at the bottom row. We have the terms 98x² and -46x. The greatest common factor of 98x² and -46x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-1	-49x	23
2x	98x²	-46x
	49x	-23

To find the values below the table, we first divide 98x² by 2x (labeled in blue). This gives us 49x.

98x² ÷ 2x = 49x

You can see this value colored in orange below:

-1	-49x	23
2x	98x²	-46x
	49x	-23

Next, we divide -46x by 2x (labeled in blue). This gives us -23.

-46x ÷ 2x = -23

You can see this value colored in purple below:

-1	-49x	23
2x	98x²	-46x
	49x	-23

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 98x² - 95x + 23. 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:

(2x - 1)(49x - 23)

That’s it! Now you know how to factor the equation 98x² - 95x + 23.

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