Factor 70x² - 99x + 29

Here we will show you how to factor the quadratic function 70x² - 99x + 29 using the box method. In other words, we will show you how to factor 70x squared minus 99x plus 29 (70x^2 - 99x + 29) 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 + 29, like this:

a = 70
b = -99
c = 29

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

-1	-70x	29
x	70x²	-29x
	70x	-29

We put 70x² (a) in the bottom left square and 29 (c) in the top right square, like this:

-1	-70x	29
x	70x²	-29x
	70x	-29

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 29 (a × c), and add together to equal -99 (b).

More specifically, 70 times 29 is 2030. Therefore, we need to find the two numbers that multiply to equal 2030, and add to equal -99.

? × ? = 2030
? + ? = -99

After looking at this problem, we can see that the two numbers that multiply together to equal 2030, and add together to equal -99, are -70 and -29, as illustrated here:

-70 × -29 = 2030
-70 + -29 = -99

Now, we can fill in the last two squares in our box with -70x and -29x. Place -70x in the upper left square, and place -29x in the lower right square.

-1	-70x	29
x	70x²	-29x
	70x	-29

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 -70x and 29. The greatest common factor of -70x and 29 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-70x	29
x	70x²	-29x
	70x	-29

Next, let’s look at the bottom row. We have the terms 70x² and -29x. The greatest common factor of 70x² and -29x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-70x	29
x	70x²	-29x
	70x	-29

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

70x² ÷ x = 70x

You can see this value colored in orange below:

-1	-70x	29
x	70x²	-29x
	70x	-29

Next, we divide -29x by x (labeled in blue). This gives us -29.

-29x ÷ x = -29

You can see this value colored in purple below:

-1	-70x	29
x	70x²	-29x
	70x	-29

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 + 29. 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)(70x - 29)

That’s it! Now you know how to factor the equation 70x² - 99x + 29.

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