Factor 4x² - 25x + 34

Here we will show you how to factor the quadratic function 4x² - 25x + 34 using the box method. In other words, we will show you how to factor 4x squared minus 25x plus 34 (4x^2 - 25x + 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 4x² - 25x + 34, like this:

a = 4
b = -25
c = 34

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

-17	-17x	34
4x	4x²	-8x
	x	-2

We put 4x² (a) in the bottom left square and 34 (c) in the top right square, like this:

-17	-17x	34
4x	4x²	-8x
	x	-2

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 4 times 34 (a × c), and add together to equal -25 (b).

More specifically, 4 times 34 is 136. Therefore, we need to find the two numbers that multiply to equal 136, and add to equal -25.

? × ? = 136
? + ? = -25

After looking at this problem, we can see that the two numbers that multiply together to equal 136, and add together to equal -25, are -17 and -8, as illustrated here:

-17 × -8 = 136
-17 + -8 = -25

Now, we can fill in the last two squares in our box with -17x and -8x. Place -17x in the upper left square, and place -8x in the lower right square.

-17	-17x	34
4x	4x²	-8x
	x	-2

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 -17x and 34. The greatest common factor of -17x and 34 is -17. Therefore, we write -17 to the left of the top row. You can see it here in the color green:

-17	-17x	34
4x	4x²	-8x
	x	-2

Next, let’s look at the bottom row. We have the terms 4x² and -8x. The greatest common factor of 4x² and -8x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-17	-17x	34
4x	4x²	-8x
	x	-2

To find the values below the table, we first divide 4x² by 4x (labeled in blue). This gives us x.

4x² ÷ 4x = x

You can see this value colored in orange below:

-17	-17x	34
4x	4x²	-8x
	x	-2

Next, we divide -8x by 4x (labeled in blue). This gives us -2.

-8x ÷ 4x = -2

You can see this value colored in purple below:

-17	-17x	34
4x	4x²	-8x
	x	-2

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 4x² - 25x + 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:

(4x - 17)(x - 2)

That’s it! Now you know how to factor the equation 4x² - 25x + 34.

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