zeros of quadratic function?

Question

1.Find a quadratic function whose zeros are 3+-sqrt of 5

2.What must be the value of k so that the graph of f(x) = x^2 +kx + 49 will intersect the x-axis exactly once?

3.Find k so that the zeros of f(x) = x^2 + (k - 3)x - 25 are additive inverses of each other.

4. Determine the nature of the roots of the quadratic equation 2x^2 + 7x - 15 = 0


ty..

Answer 1

1. (x-3+sqrt5)(x-3-sqrt5) = (x-3)^3 - 5 = x^2 - 6x + 9 - 5
f(x) = x^2 - 6x + 4

2. it should be a perfect square trinomial: k = 14 or -14.

3. middle term coefficient is the negative of the sum of the roots.
k - 3 = 0 then k = 3.

4. discriminant: 49 - 4(2)(-15) is positive... then the roots are real and unequal.


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Answer 2

1. (x^2)/2 - 3x + 2 OR x^2 - 6x + 4.
Any function k(x^2 - 6x + 4), where k is a positive integer will intersect the x-axis at 3(+/-)sqrt(5), as you've required.

2. f(x) = x^2 +kx + 49 (at any zero, f(x) = 0)

So if f(x) = 0 at x = 1, then we can substitute these into the function as follows:

0 = 1^2 + k + 49
0 = 1 + k +49
0 = 50 + k so therefore k must be -50

Hence, f(x) = x^2 - 50x + 49

3. k = 3 so f(x) = x^2 - 25 where if f(x) = 0 then x = (+/-)5.

-5 + 5 = 0 so this satisfies itself as an additive inverse.

4. I don't want to be repetitive for this question as the others have answered this well. Please refer to the previous answers for Question 4.

I hope this helps and good luck!

Answer 3

1. [x - (3 + sqrt(5))]*[x - (3 - sqrt(5))] = 0. The expansion is easier if you notice that the equation is
[(x - 3) - sqrt(5)]*[(x - 3) + sqrt(5)] = (x - 3)^2 - (sqrt(5)^2 = 0.

2. The graph will intersect the x-axis only once when the vertex is on the x-axis. Since the x-coordinate of the vertex satisfies 2x + k = 0, you should be able to finish this.

3. Use the quadratic formula; find roots a and b, and solve
a + b = 0.

4. Use the discriminant D = b^2 - 4ac. The roots are real and distinct if D > 0, real and equal if D = 0, and complex (conjugates) if D < 0.

Answer 4

Question Number 1 : For this equation x^2 + 5*x + 4 = 0 , answer the following questions : A. Use factorization to find the root of the equation ! Answer Number 1 : The equation x^2 + 5*x + 4 = 0 is already in a*x^2+b*x+c=0 form. In that form, we can easily derive that the value of a = 1, b = 5, c = 4. 1A. Use factorization to find the root of the equation ! x^2 + 5*x + 4 = 0 <=> ( x + 1 ) * ( x + 4 ) = 0 So we got the answers as x1 = -1 and x2 = -4 so , none of them is the answer

Answer 5

1. x^2 -8x+15=F(x)