Factor -3x² + 100x

Factor -3x² + 100x - 97

Here we will show you how to factor the quadratic function -3x² + 100x - 97 using the box method. In other words, we will show you how to factor negative 3x squared plus 100x minus 97 (-3x^2 + 100x - 97) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 3x² - 100x + 97. Now we can label the different parts of our equation, like this:

a = 3
b = -100
c = 97

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

-97	-97x	97
3x	3x²	-3x
	x	-1

We put 3x² (a) in the bottom left square and 97 (c) in the top right square, like this:

-97	-97x	97
3x	3x²	-3x
	x	-1

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 3 times 97 (a × c), and add together to equal -100 (b).

More specifically, 3 times 97 is 291. Therefore, we need to find the two numbers that multiply to equal 291, and add to equal -100.

? × ? = 291
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 291, and add together to equal -100, are -97 and -3, as illustrated here:

-97 × -3 = 291
-97 + -3 = -100

Now, we can fill in the last two squares in our box with -97x and -3x. Place -97x in the upper left square, and place -3x in the lower right square.

-97	-97x	97
3x	3x²	-3x
	x	-1

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 -97x and 97. The greatest common factor of -97x and 97 is -97. Therefore, we write -97 to the left of the top row. You can see it here in the color green:

-97	-97x	97
3x	3x²	-3x
	x	-1

Next, let’s look at the bottom row. We have the terms 3x² and -3x. The greatest common factor of 3x² and -3x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

-97	-97x	97
3x	3x²	-3x
	x	-1

To find the values below the table, we first divide 3x² by 3x (labeled in blue). This gives us x.

3x² ÷ 3x = x

You can see this value colored in orange below:

-97	-97x	97
3x	3x²	-3x
	x	-1

Next, we divide -3x by 3x (labeled in blue). This gives us -1.

-3x ÷ 3x = -1

You can see this value colored in purple below:

-97	-97x	97
3x	3x²	-3x
	x	-1

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 -3x² + 100x - 97. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(3x - 97)(x - 1)

In our original quadratic equation, -3x² + 100x - 97, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x - 97)(x - 1)

That’s it! Now you know how to factor the equation -3x² + 100x - 97.

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