Factor 61x² - 95x + 34

Here we will show you how to factor the quadratic function 61x² - 95x + 34 using the box method. In other words, we will show you how to factor 61x squared minus 95x plus 34 (61x^2 - 95x + 34) 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 61x² - 95x + 34, like this:

a = 61
b = -95
c = 34

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

-1	-61x	34
x	61x²	-34x
	61x	-34

We put 61x² (a) in the bottom left square and 34 (c) in the top right square, like this:

-1	-61x	34
x	61x²	-34x
	61x	-34

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 61 times 34 (a × c), and add together to equal -95 (b).

More specifically, 61 times 34 is 2074. Therefore, we need to find the two numbers that multiply to equal 2074, and add to equal -95.

? × ? = 2074
? + ? = -95

After looking at this problem, we can see that the two numbers that multiply together to equal 2074, and add together to equal -95, are -61 and -34, as illustrated here:

-61 × -34 = 2074
-61 + -34 = -95

Now, we can fill in the last two squares in our box with -61x and -34x. Place -61x in the upper left square, and place -34x in the lower right square.

-1	-61x	34
x	61x²	-34x
	61x	-34

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 -61x and 34. The greatest common factor of -61x and 34 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-61x	34
x	61x²	-34x
	61x	-34

Next, let’s look at the bottom row. We have the terms 61x² and -34x. The greatest common factor of 61x² and -34x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-61x	34
x	61x²	-34x
	61x	-34

To find the values below the table, we first divide 61x² by x (labeled in blue). This gives us 61x.

61x² ÷ x = 61x

You can see this value colored in orange below:

-1	-61x	34
x	61x²	-34x
	61x	-34

Next, we divide -34x by x (labeled in blue). This gives us -34.

-34x ÷ x = -34

You can see this value colored in purple below:

-1	-61x	34
x	61x²	-34x
	61x	-34

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 61x² - 95x + 34. 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 - 1)(61x - 34)

That’s it! Now you know how to factor the equation 61x² - 95x + 34.

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