Quadratic to Binomials

[freemathhelpuser]

A lesson just taught that when a quadraic is x^2+ (a constant), we can find the square factors of that constant and put them into the equation. An example is below.

The rectangle below has an area of [FONT=KaTeX_Main]x^2-16
x, start superscript, 2, end superscript, minus, 16 x^2-16 square meters and a width of [FONT=KaTeX_Main]x+4[FONT=KaTeX_Math]x+4x, plus, 4[/FONT] meters.[/FONT]
[/FONT]
What expression represents the length of the rectangle?

[FONT=KaTeX_Main]\text{Length} =Length= x-4 meters

We learn here that x^2-16 = (x-4)*(x+4)

This is not always correct, however:

x^2+144 does not equal (x+12)*(x+12) even though we took the square root of 144. (x+12)*(x+12) = X^2+24x+144

What did I not get from the lesson?
L, e, n, g, t, h, equal
[/FONT]

 

[ksdhart2]

...when a quadratic is x^2 + (a constant), we can find the square factors of that constant and put them into the equation.

This is not true at all, not for any value. Generally speaking, a polynomial of the form x²+0x+c, where c is any real number, cannot be factored. There is a way around this which deals with a new type of a number, called complex numbers, but that's a topic for a much more advanced math course. For now, it's sufficient to say that those polynomials are impossible to factor.

However, if you replace the plus with a minus sign, you get an expression which can always be factored. Specifically: x²+0x-c, where c is any real number, will factor to \(\displaystyle \left(x-\sqrt{c}\right)\left(x+\sqrt{c}\right)\). This works best for mental arithmetic if c is a perfect square, but the premise holds for values of c that are not perfect squares.

 

[stapel]

A lesson just taught that when a quadraic is x^2+ (a constant), we can find the square factors of that constant and put them into the equation. An example is...x^2-16

But this is not "x² + (a constant)"; it's "x² - (a constant)". That change, from "plus" to "minus", makes a huge difference!

Since you've posted this to "Pre-Algebra", I'm going to guess that maybe something was "lost in translation" when attempting to share something that you'll see once you finally reach algebra classes. Short version:

. . . . .\(\displaystyle x^2\, -\, a\, =\, \left(x\, -\, \sqrt{\strut a\,}\right)\,\left(x\, +\, \sqrt{\strut a\,}\right), \mbox{ with }\, a\, \geq\, 0\)

. . . . .\(\displaystyle x^2\, +\, a\, \mbox{ cannot be factored using any numbers you've ever used before}\)