Factoring with Grouping 03:52 minutes

Video Transcript

Transcript Factoring with Grouping

Meet Vincent. He’s a painter and quite the eccentric. He just got a new commission, to embellish the façade of an old building on the fancy side of the town. The neighbors just want the ugly wall to disappear. For inspiration, he goes to take a look at the building. Those darn kids – they painted graffiti all over it. No matter, Vincent will paint over the mess. To help him cover the graffiti, Vincent can solve quadratic equations by factoring and grouping.

Back at home, as he considers the specifics of the building, he divises a strategy. What does he know? The front of the building is equal to an area of 45 yards squared, but not to be included in the painting is the fire escape on the right side of the building, it's 2 yards wide, and the windows at the top of the building, they have a height of 3 yards. The painting must be in the shape of a rectangle with the height equal to 2 times the width. Because he doesn’t know the width or the height, he uses the variables, x and 2x.

Setting up an equation

He sets up an equation and sets the total area equal to 45 yards squared. The quantity x plus 2 times the quantity 2x plus 3 is equal to 45. To calculate x, first we FOIL. Then, we combine like terms. Next, since this is a quadratic equation, to find the solution, we modify the equation - so it's equal to 0.

Factoring by Grouping

Now we're ready to factor with grouping. First, find the factors of ac that sum to b. ac is equal to 2 times -39, so -78. b is 7. Now what factors of -78 sum to 7? Hmmm, let’s go through the list. Ahah, -6 and 13 will work. Now, we split up 7x into two terms, -6x and 13x. Pay close attention to this next step: use parentheses to group the 4 terms into 2 binomials and then factor out the GCF from each binomial. This can be tricky, so watch carefully. The end result is the binomial: x - 3 times the binomial 2x + 13, and the product is equal to zero. But not so fast, we have one last step.
Apply the Zero Product Property and solve for both values of 'x'. There one last thing to think about with this problem. You can’t have a negative length or width, so only one of the solutions is a possible answer. X is equal to 3 yards, so the height of the painting is equal to 6 yards and the width is equal to 3 yards.

Now that the math is done, Vincent can work on his masterpiece…