Factor 70x² - 99x + 35

Here we will show you how to factor the quadratic function 70x² - 99x + 35 using the box method. In other words, we will show you how to factor 70x squared minus 99x plus 35 (70x^2 - 99x + 35) 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 70x² - 99x + 35, like this:

a = 70
b = -99
c = 35

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

-5	-50x	35
7x	70x²	-49x
	10x	-7

We put 70x² (a) in the bottom left square and 35 (c) in the top right square, like this:

-5	-50x	35
7x	70x²	-49x
	10x	-7

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 70 times 35 (a × c), and add together to equal -99 (b).

More specifically, 70 times 35 is 2450. Therefore, we need to find the two numbers that multiply to equal 2450, and add to equal -99.

? × ? = 2450
? + ? = -99

After looking at this problem, we can see that the two numbers that multiply together to equal 2450, and add together to equal -99, are -50 and -49, as illustrated here:

-50 × -49 = 2450
-50 + -49 = -99

Now, we can fill in the last two squares in our box with -50x and -49x. Place -50x in the upper left square, and place -49x in the lower right square.

-5	-50x	35
7x	70x²	-49x
	10x	-7

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 -50x and 35. The greatest common factor of -50x and 35 is -5. Therefore, we write -5 to the left of the top row. You can see it here in the color green:

-5	-50x	35
7x	70x²	-49x
	10x	-7

Next, let’s look at the bottom row. We have the terms 70x² and -49x. The greatest common factor of 70x² and -49x is 7x. Therefore, we write 7x to the left of the bottom row. You can see it here in the color blue:

-5	-50x	35
7x	70x²	-49x
	10x	-7

To find the values below the table, we first divide 70x² by 7x (labeled in blue). This gives us 10x.

70x² ÷ 7x = 10x

You can see this value colored in orange below:

-5	-50x	35
7x	70x²	-49x
	10x	-7

Next, we divide -49x by 7x (labeled in blue). This gives us -7.

-49x ÷ 7x = -7

You can see this value colored in purple below:

-5	-50x	35
7x	70x²	-49x
	10x	-7

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 70x² - 99x + 35. 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:

(7x - 5)(10x - 7)

That’s it! Now you know how to factor the equation 70x² - 99x + 35.

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