Solving Quadratic Equations

[1038373daj]

I read through the instructional post, however i do not even know how to begin these three questions, because i missed lecture so i would really appreciate your help!

Problem # 1:

(c-12)c-15=30

Answer: c = 3 or c = 15

Next, Problem # 2:

9 + (r-5) r = 33

Answer r=8 or r= -3 ??

And last question of the day,

Problem #3:

(t + 4) (t+7)=54

Answer: t= -13 or t=2

If you could help explain step by step how these problems are solved, i have other problems like these in the homework i just am stuck knowing where to begin!
Thanks for any help you can give!
Sincerely, Danielle

 

[BigGlenntheHeavy]

\(\displaystyle All \ right \ Danielle, \ I'll \ do \ the \ first \ one \ for \ you \ to \ get \ you \ started.\)

\(\displaystyle (c-12)c-15 \ = \ 30, \ distributing \ the \ c \ gives: \ c^2-12c-15 \ = \ 30\)

\(\displaystyle c^2-12c-15-30 \ = \ 0, \ c^2-12c-45 \ = \ 0, \ (c-15)(c+3) \ = \ 0\)

\(\displaystyle Hence, \ c \ = \ 15 \ or \ c \ = \ -3, \ you \ do \ the \ check.\)

 

[mmm4444bot]

(c-12)c-15=30

Answer: c = 3 or c = 15

c = 3 is not a solution.

Substituting 3 for c in the given equation leads to -42 = 30.

Here are the steps, for each of these exercises (plus an example, at the end).

Multiply everything, then collect all terms to one side and simplify.

You will end up with an equation in this form:

Ax^2 + Bx + C = 0

where A, B, and C are integers.

Now factor the polynomial by finding two integers whose product is C and whose sum is B.

Finish by setting each factor equal to zero, and solve for the variable.

EG:

(x - 9)x = 22

Multiply everything.

x^2 - 9x = 22

Collect terms to one side.

x^2 - 9x - 22 = 0

A = 1
B = -9
C = -22

Find two numbers whose product is -22 and whose sum is -9.

These numbers are 2 and -11, so the factorization is:

(x + 2)(x - 11) = 0

Set each factor equal to zero, and solve for x.

x + 2 = 0

x - 11 = 0

The solutions are x = -2 or x = 11.