SOLUTION: can you help me with these problems?? thank you so much 1.) {{{a^2-4a+3=0}}} 2.) {{{x^2-4x-10=0}}} 3.) {{{3b^2+4b-2=0}}} 4.) {{{2c^2+5c-2=0}}}

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: can you help me with these problems?? thank you so much 1.) {{{a^2-4a+3=0}}} 2.) {{{x^2-4x-10=0}}} 3.) {{{3b^2+4b-2=0}}} 4.) {{{2c^2+5c-2=0}}}      Log On


   

Question 197253: can you help me with these problems?? thank you so much
1.) a%5E2-4a%2B3=0
2.) x%5E2-4x-10=0
3.) 3b%5E2%2B4b-2=0
4.) 2c%5E2%2B5c-2=0
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started...


# 1

a%5E2-4a%2B3=0 Start with the given equation.


Notice we have a quadratic in the form of Aa%5E2%2BBa%2BC where A=1, B=-4, and C=3


Let's use the quadratic formula to solve for a


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


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


a+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%283%29+%29%29%2F%282%281%29%29 Negate -4 to get 4.


a+=+%284+%2B-+sqrt%28+16-4%281%29%283%29+%29%29%2F%282%281%29%29 Square -4 to get 16.


a+=+%284+%2B-+sqrt%28+16-12+%29%29%2F%282%281%29%29 Multiply 4%281%29%283%29 to get 12


a+=+%284+%2B-+sqrt%28+4+%29%29%2F%282%281%29%29 Subtract 12 from 16 to get 4


a+=+%284+%2B-+sqrt%28+4+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


a+=+%284+%2B-+2%29%2F%282%29 Take the square root of 4 to get 2.


a+=+%284+%2B+2%29%2F%282%29 or a+=+%284+-+2%29%2F%282%29 Break up the expression.


a+=+%286%29%2F%282%29 or a+=++%282%29%2F%282%29 Combine like terms.


a+=+3 or a+=+1 Simplify.


So the solutions are a+=+3 or a+=+1




# 2


x%5E2-4x-10=0 Start with the given equation.


Notice we have a quadratic in the form of Ax%5E2%2BBx%2BC where A=1, B=-4, and C=-10


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-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%28-10%29+%29%29%2F%282%281%29%29 Plug in A=1, B=-4, and C=-10


x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%28-10%29+%29%29%2F%282%281%29%29 Negate -4 to get 4.


x+=+%284+%2B-+sqrt%28+16-4%281%29%28-10%29+%29%29%2F%282%281%29%29 Square -4 to get 16.


x+=+%284+%2B-+sqrt%28+16--40+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-10%29 to get -40


x+=+%284+%2B-+sqrt%28+16%2B40+%29%29%2F%282%281%29%29 Rewrite sqrt%2816--40%29 as sqrt%2816%2B40%29


x+=+%284+%2B-+sqrt%28+56+%29%29%2F%282%281%29%29 Add 16 to 40 to get 56


x+=+%284+%2B-+sqrt%28+56+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%284+%2B-+2%2Asqrt%2814%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%284%29%2F%282%29+%2B-+%282%2Asqrt%2814%29%29%2F%282%29 Break up the fraction.


x+=+2+%2B-+sqrt%2814%29 Reduce.


x+=+2%2Bsqrt%2814%29 or x+=+2-sqrt%2814%29 Break up the expression.


So the solutions are x+=+2%2Bsqrt%2814%29 or x+=+2-sqrt%2814%29


which approximate to x=5.742 or x=-1.742


Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


1.)

This factors because and , so:



Use the Zero Product Rule:



or



2.) does not factor.

You can tell whether a quadratic will factor by computing the discriminant. The discriminant is the expression under the radical in the quadratic formula, namely . If the result is not a perfect square, then the quadratic does not factor over the rationals. Here you have -4 squared is 16 and 4 times 1 times -10 = -40 and 16 - (-40) is 56 which is not a perfect square.

Since the quadratic does not factor, you can either complete the square or use the much easier process of using the quadratic formula:





The other two do not factor either, proof of which is left as an exercise for the student. You do them the same way -- use the quadratic formula.

John