Question

Which of the following tables shows the correct steps to transform x^2 + 10x + 24 = 0 into the form (x - p)^2 = q?
[p and q are integers]

Answer 1

@jdoe0001

Answer 2

do you know what a perfect square trinomial is?

Answer 3

Isn't that where you can't factor it more?

Answer 4

well... not quite..... a perfect square trinomial would be \(\bf a^2\pm2ab+b^2\iff (a\pm b)^2\\
\qquad \uparrow \\
\textit{perfect square trinomial}\)

Answer 5

TRInomial, tri=three terms

Answer 6

lemme give you a quick example \(\bf (3x+2y)^2\implies (3x)^2+2\cdot 3x\cdot 2y+(2y)^2\)

see the correspondence to the squared binomial?

Answer 7

i see it would the answer be D

Answer 8

well... what makes you think is D?

Answer 9

what makes me think D is because i know that a and b are wrong because they don't use the same equation from the question and i don't think it is C because of teh double parentheses so by that process it makes me think D

Answer 10

*the

Answer 11

@jdoe0001

Answer 12

I think the parentheses is just a typo
and they all use the same equation to start off from

Answer 13

oh wait sorry i just realized that my reasoning is wrong when i looked at it again

Answer 14

Did i get it right anyway with D?

Answer 15

so let's try to "complete the square"
\(\bf x^2 + 10x + 24 = 0\implies x^2 + 10x =-24\implies (x^2 + 10x) =-24
\\ \quad \\
(x^2 + 10x+{\color{red}{ \square }}^2) =-24\)

so what do you think would be a number that would "complete the " perfect square trinomial?

Answer 16

recall \(\bf \Large \bf a^2\pm2ab+b^2\iff (a\pm b)^2\)

Answer 17

and that our perfect square trinomial is \(\bf \Large \bf a^2\pm2ab+b^2\)

Answer 18

so 2?

Answer 19

well. 2 there is just part of the template
not a value for "a" or "b"

Answer 20

10?

Answer 21

well, 2 * a * b
or
2 * first term * 2nd term = 10x

we know the first term, is "x"
we dunno the 2nd term

but we know that
2 * first term * 2nd term = 10x
2 * x * 2nd term = 10x

so \(\bf 2\cdot x\cdot {\color{red}{ \square }}=10x\) so what do you think is our 2nd term?

Answer 22

24

Answer 23

24 ok let's see

2 * x * 24 = 10x? well

2 * x * 24 = 2 * 24 * x => 48x

so.. is not 24 though

Answer 24

5

Answer 25

2 * x * 5 = 10x

Answer 26

5 let's try that one
2 * x * 5 = 10?

2 * x * 5 => 2 * 5* x => 10x yes

so now we know that our 2nd term is 5 in the perfect square trinomial
thus

we would ADD that to the equation

keep in mind that all we're doing is borrowing from mr zero, 0, our good friend

so if we ADD \(5^2\) we have to also SUBTRACT \(5^2\)

thus \(\bf x^2 + 10x + 24 = 0\implies x^2 + 10x =-24\implies (x^2 + 10x) =-24
\\ \quad \\
(x^2 + 10x+{\color{red}{ 5 }}^2)-{\color{red}{ 5 }}^2 =-24\implies (x+5)^2-25=-24
\\ \quad \\
(x+5)^2=-24+25\implies (x+5)^2=1\)

Answer 27

so that means that the answer is d right?

Answer 28

notice that to "complete the square" all we really needed was 25
and we started off with 24

so we really could have just added +1 :)
\(\bf x^2 + 10x + 24 = 0
\\ \quad \\
x^2 + 10x + 24 +1= 0+1
\\ \quad \\
x^2 + 10x +25=1\implies (x+5)^2=1\)

yeap

Answer 29

thanks

Answer 30

yw