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Course: Get ready for Algebra 2>Unit 1

Lesson 9: Factoring quadratics with perfect squares

Identifying perfect square form

Sal shows how we can identify that a trinomial has the "perfect square" form.

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• Can the answer also be negative?(-5x-2)?
• If you mean (-5x-2)^2, then yes, but -5x-2 is not the same.
• Is possible to factor this quadratic:
25x^2 + 16x + 9 as (a + b)^2 or in anyway?
• No it wouldn't work.
Yes... your quadratic has perfect squares at both ends, but the middle term is incorrect.
The terms on the ends of your factors would need to be: (5x + 3)^2
If you multiply this out, you get: 25x^2 + 30x + 9

Hope this helps.
• How is Sal so good at writing with a mouse?
• He uses a stylus (like a pen).
• When in life am I ever going to use this stuff?
• ya fr
• This seems sort of complicated... Why don't we just factor the binomial like we usually would? Like for example, when Sal had 25x^2 + 20x + 4, I did it normally and got the same answer that he did, (5x+2)^2.
• Why is it (Ax)^2 and not just Ax^2? Sorry and thank you to whoever answers! :)
(1 vote)
• (Ax)² = (Ax) * (Ax)
Ax² = A * x² = A * x * x
• At the numbers in the brackets are multiplied by each other how does this work?
• I'm a bit confused about the second term in the perfect square form.

In this video, Sal said that 20x (the center term of 25x^2 + 20x + 4) is 2.A.B.x, implying that A is the coefficient of 25^2 and x was just added on because that's how the formula works.

However, in the "Practice: Perfect Squares" exercise, a solution says that 14x (the center term of x^2 + 14x + 49) is twice the root of 1x^2 and 49, implying that A is the coefficiant AND the variable of 1x^2.

Can someone please explain which one is correct, if both are, or if I messed something up? I'm not sure if this matters much, but it confused me when I got to Completing the Square in the next unit.
• Consider what happens when you multiply: (ax+b)(ax+b)
You get: (ax)^2+abx+abx+b^2 which simplifies to a^2x^2+2ab+b^2
The "a" refers to the square root of the coefficient of the x^2 term. The "b" refers to the square root of the constant term.
For 25x^2 + 20x + 4: a=5 and b=2. 2ab = 2(5)(2) = 20
For x^2 + 14x + 49: a=1 and b=7. 2ab = 2(1)(7) = 7

Hope this helps.
• Hello, I'm studying for an exam and our workbook has this question:

The product of two numbers is 120, and the sum of their squares is 289. The sum of the number is __."

Does anyone know a course/video that talks about these types of problems? Or does anyone know how to solve this? I know what the answer is but I don't really understand how this works.
(1 vote)
• Use X = one number and Y = 2nd number
Translate the first part of the 1st sentence and you get:
xy=120
Translate the second part of the 1st sentence and you get:
x^2+y^2=289

You now have a system of equations.
-- Solve the 1st equation for one of the 2 variables by dividing by sides: y=120/x
-- Substitute this into the 2nd equation in place of Y to get: x^2+(120/x)^2 = 289
-- Do the exponent: x^2+14400/x^2 = 289
-- Multiply both sides by x^2: x^4 + 14400 = 289x^2
-- Subtract 289x^2 from both sides: x^4-289x^2+14400 = 0
-- Factor: (x^2-225)(x^2-64) = 0
-- Factor more: (x-15)(x+15)(x-8)(x+8)=0
-- Split the factors apart and solve each.
x-15=0 creates x=15
x+15=0 creates x=-15
x-8=0 creates x=8
x+8=0 creates x=-8

To find y: use y=120/x
x=15 creates y=120/15 = 8
x=-15 creates y=120/(-15) = -8
x=8 creates y=120/8 = 15
x=-8 creates y=120/(-8) = -15
Basically, the two numbers are either 8 and 15, or -8 and -15.

Hope this helps.
• To all fellow students asking why we have to learn this and where we will use this:

"Academic" has a few definitions. One of them is "relating to education". Another is "lacking practical relevance; of theoretical interest only".

Relevance is a puzzle. It is completely dependent on a goal---as Booker T. Washington said, all successes in life started out with a goal in mind.

So, education lacks practical relevance based on our goals.

If you want a scientific career, you will need to know all this. It will have "practical relevance".
If you want to work for a grocery store, this can all be of "theoretical interest only"!