SOLUTION: Complete parts a-c for each quadratic equation a) find the value of the discriminant b) describe the number and type of roots c) find the exact solutions by using the Quadrat

Algebra ->  Rational-functions -> SOLUTION: Complete parts a-c for each quadratic equation a) find the value of the discriminant b) describe the number and type of roots c) find the exact solutions by using the Quadrat      Log On


   

Question 181944This question is from textbook Algebra2
: Complete parts a-c for each quadratic equation
a) find the value of the discriminant
b) describe the number and type of roots
c) find the exact solutions by using the Quadratic Formula
14) x^2+3x-3=0 This question is from textbook Algebra2

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)


From x%5E2%2B3x-3=0 we can see that a=1, b=3, and c=-3


D=b%5E2-4ac Start with the discriminant formula.


D=%283%29%5E2-4%281%29%28-3%29 Plug in a=1, b=3, and c=-3


D=9-4%281%29%28-3%29 Square 3 to get 9


D=9--12 Multiply 4%281%29%28-3%29 to get %284%29%28-3%29=-12


D=9%2B12 Rewrite D=9--12 as D=9%2B12


D=21 Add 9 to 12 to get 21

So the discriminant is 21.
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b)

From part a), we see that D=21. This means that D%3E0 (ie the discriminant is positive)

Since the discriminant is greater than zero, this means that there are two real solutions.

So we can say that...

Type of Solution(s): Real
Number: 2 (these are distinct)


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c)



x%5E2%2B3x-3=0 Start with the given equation.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=3, and c=-3


Let's use the quadratic formula to solve for x


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


x+=+%28-%283%29+%2B-+sqrt%28+%283%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29 Plug in a=1, b=3, and c=-3


x+=+%28-3+%2B-+sqrt%28+9-4%281%29%28-3%29+%29%29%2F%282%281%29%29 Square 3 to get 9.


x+=+%28-3+%2B-+sqrt%28+9--12+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-3%29 to get -12


x+=+%28-3+%2B-+sqrt%28+9%2B12+%29%29%2F%282%281%29%29 Rewrite sqrt%289--12%29 as sqrt%289%2B12%29


x+=+%28-3+%2B-+sqrt%28+21+%29%29%2F%282%281%29%29 Add 9 to 12 to get 21


x+=+%28-3+%2B-+sqrt%28+21+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-3%2Bsqrt%2821%29%29%2F%282%29 or x+=+%28-3-sqrt%2821%29%29%2F%282%29 Break up the expression.


So the answers are x+=+%28-3%2Bsqrt%2821%29%29%2F%282%29 or x+=+%28-3-sqrt%2821%29%29%2F%282%29


which approximate to x=0.791 or x=-3.791


Notice how there are two real solutions. So this confirms our answer to part b)