SOLUTION: How do you solve this quadratic equation??? x squared -17x-9=0

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: How do you solve this quadratic equation??? x squared -17x-9=0      Log On


   

Question 271137: How do you solve this quadratic equation???
x squared -17x-9=0
Found 2 solutions by Greenfinch, Theo:
Answer by Greenfinch(383) About Me  (Show Source):
You can put this solution on YOUR website!
x^2 - 17x - 9 = 0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-17x%2B-9+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-17%29%5E2-4%2A1%2A-9=325.

Discriminant d=325 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--17%2B-sqrt%28+325+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-17%29%2Bsqrt%28+325+%29%29%2F2%5C1+=+17.51387818866
x%5B2%5D+=+%28-%28-17%29-sqrt%28+325+%29%29%2F2%5C1+=+-0.513878188659973

Quadratic expression 1x%5E2%2B-17x%2B-9 can be factored:
1x%5E2%2B-17x%2B-9+=+1%28x-17.51387818866%29%2A%28x--0.513878188659973%29
Again, the answer is: 17.51387818866, -0.513878188659973. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-17%2Ax%2B-9+%29

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
equation is x^2 - 17x - 9 = 0

it cannot be factored without using the quadratic formula or the completing the squares method.

quadratic formula method is:

x = quadratic formula is %28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29

equation has to be in standard form of ax^2 + bx + c = 0 which it is.

a = 1
b = -17
c = -9

substitute in quadratic formula to get:

x = %28-%28-17%29+%2B-+sqrt%28%28-17%29%5E2-4%2A1%2A%28-9%29%29%29%2F%282%2A1%29

you get:

x = %2817+%2B-+sqrt%28289%2B36%29%29%2F2 which becomes:

x = %2817+%2B-+sqrt%28325%29%29%2F2 which becomes:

x = %2817+%2B-+18.02775638%29%2F2 which becomes:

x = 17.51387819
or
x = -.0513878189

substitute in your original equation to confirm the answers are good.

your original equation is:

x^2 - 17x - 9 = 0

substituting 17.5387819 for x gets 0 = 0 so the first answer is good.

substituting -.5138978189 for x gets 0 = 0 so the second answer is good.

those are your factors.

completing the squares method works as follows:

your original equation is

x^2 - 17x - 9 = 0

add 9 to both sides of the equation to get

x^2 - 17x = 9

take 1/2 of the b term to get:

b/2 = -17/2 = -8.5

square that to get (b/2)^2 = 72.25

your completing the squares formula is:

%28x-8.5%29%5E2+=+9+%2B+72.25+=+81.25

take the square root of both sides of this equation to get:

x+-+8.5 = +/- sqrt%2881.25%29

add 8.5 go both sides of this equation to get:

x+=+8.5 +/- sqrt%2881.25%29

solve for x to get:

x = 17.51387819
or

x = -.513878189

they're the same answers you got using the quadratic formula.

you should remember the quadratic formula at least.

anytime you get a quadratic equation that you can't factor easily, use the quadratic formula.

if you prefer to use the completing the squares method, then remember that.

it's a little harder to explain how that works, but it's not too bad to work with once you know how.

you need to remember at least one of them.

the quadratic formula is a formula.

the completing the square method is a method.

i believe the quadratic formula can be derived from the completing the squares method.

while the factoring method only works on quadratic equations when the factors are integers, the quadratic equation and the completing the squares method work on any quadratic equation.

even when the factoring method can be used, it's not always easy to spot the factors, especially when the a term (coefficient of the x^2 term) is not equal to 1.

bottom line is:

don't waste time trying to find the factors using the factoring method unless the question absolutely requires that you solve it that way.

as soon as you are having difficulty finding the factors, revert to the quadratic equation or the completing the squares method, whichever you feel more comfortable with.

here's some references for both the quadratic formula and the completing the squares method.

http://www.purplemath.com/modules/sqrquad.htm
http://www.purplemath.com/modules/quadform.htm
http://www.jamesbrennan.org/algebra/quadratics/the_quadratic_formula.htm