Factor 30x² + x

Factor 30x² + x - 69

Here we will show you how to factor the quadratic function 30x² + x - 69 using the box method. In other words, we will show you how to factor 30x squared plus x minus 69 (30x^2 + x - 69) 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 30x² + x - 69, like this:

a = 30
b = 1
c = -69

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

-3	-45x	-69
2x	30x²	46x
	15x	23

We put 30x² (a) in the bottom left square and -69 (c) in the top right square, like this:

-3	-45x	-69
2x	30x²	46x
	15x	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 30 times -69 (a × c), and add together to equal 1 (b).

More specifically, 30 times -69 is -2070. Therefore, we need to find the two numbers that multiply to equal -2070, and add to equal 1.

? × ? = -2070
? + ? = 1

After looking at this problem, we can see that the two numbers that multiply together to equal -2070, and add together to equal 1, are -45 and 46, as illustrated here:

-45 × 46 = -2070
-45 + 46 = 1

Now, we can fill in the last two squares in our box with -45x and 46x. Place -45x in the upper left square, and place 46x in the lower right square.

-3	-45x	-69
2x	30x²	46x
	15x	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 -45x and -69. The greatest common factor of -45x and -69 is -3. Therefore, we write -3 to the left of the top row. You can see it here in the color green:

-3	-45x	-69
2x	30x²	46x
	15x	23

Next, let’s look at the bottom row. We have the terms 30x² and 46x. The greatest common factor of 30x² and 46x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-3	-45x	-69
2x	30x²	46x
	15x	23

To find the values below the table, we first divide 30x² by 2x (labeled in blue). This gives us 15x.

30x² ÷ 2x = 15x

You can see this value colored in orange below:

-3	-45x	-69
2x	30x²	46x
	15x	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:

-3	-45x	-69
2x	30x²	46x
	15x	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 30x² + x - 69. 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 - 3)(15x + 23)

That’s it! Now you know how to factor the equation 30x² + x - 69.

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Factor 30x² + x - 66
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