If 4x^2+kx=-9, find the value of k, k>0 such that the equation has two equal roots.?

Answer 1

4x^2 + kx + 9 = 0.
Two methods: a) there are two equal roots if the discriminant, which is b^2 - 4ac, equals 0.
So k^2 - 4*4*9 = 0, so k = 12. (since k>0)
b) there are two equal roots if the quadratic is actually a perfect square. You need a 4 = 2^2 on the front, and a 3 = 3^2 on the end, so it must equal (2x+3)^2 (since the middle bit must be > 0).That is 4x^2 + 12x + 9, so k = 12.

Answer 2

4x² + kx = -9

Transpose -9
4x² + kx + 9 = 0

We then find the discriminant. This is how it works:

__________________________
The Discriminant
If the quadratic equation is
ax² + bx + c = 0

Then the value of the discriminant is
b² - 4ac

Now,

a) If the discriminant is positive (or b² - 4ac > 0) then the equation has 2 distinct and real roots.

b) If the discriminant is zero (or b² - 4ac = 0) then the equation has 2 real and equal roots.

c) If the discriminant is negative (or b² - 4ac < 0) then the equation has 2 imaginary roots.
_______________________
Now, from your equation
4x² + kx + 9 = 0

we can find the value of a, b and c
a = 4, b = k and c = 9

Since we need 2 equal roots, then we need the discriminant to be equal to zero, or
b² - 4ac = 0

Now, substitute the values of a, b and c
k² - 4(4)(9) = 0

Then
k² - 144 = 0

Factor out
(k - 12)(k + 12) = 0

Therefore,
k = 12 or k = -12

Since you want k > 0, then the only acceptable value for k is
k = 12

^_^

Answer 3

Add 9 to both sides to get
4x^2 + kx + 9 = 0

Having equal roots means if you factorise it you get (.....)^2

Now to get 4x^2 you have to square 2x; to get 9 you square +or- 3.
What do you get if you expand (2x +or- 3)^2? The first term is 4x^2, the third one is 9, and the middle one should tell you what k is.

Another way, but not quite so informative, is the formal rule that a quadratic equation has equal roots when the discriminant, b^2-4ac, is 0. In this case b = k, a = 4 and c = 9, so you can solve that to find the two possible values of k.

Answer 4

This is 4x^2 + kx + 9 =0. The quadratic equation has equal roots when its discriminant b^2-4ac is zero. In this case a=4, b=k and c=9. So

k^2 - 4*4*9 = 0
k^2 = 12^2
k=±12. Since k>0 we keep k=+12.

Answer 5

Completing the square:
4 x^2 + k x + 9
= 4 (x^2 + (k/4) x + 9/4)
= 4 (x^2 + 2 (k/8) x + (k/8)^2 - (k/8)^2 + 9/4)
= 4 (x + k/8)^2 - 4 (k^2/64) + 9
= 4 (x + k/8)^2 - k^2/16 + 9
= 0
Thus, solving for x:
4 (x + k/8)^2 = k^2/16 - 9
(x + k/8)^2 = k^2/64 - 9/4
For this to have two equal roots, we must have
k^2/64 - 9/4 = 0,
or
k^2/64 = 9/4
k^2 =64*9/4 = 144
Therefore, k = 12 or k = -12. Since k > 0, we have k = 12.

Answer 6

Rewriting the given eqn:
4x^2 + kx + 9 = 0

This is of the form: ax^2 + bx + c = 0
The roots of the equation are: [-b +/- sqrt(b^2 - 4ac)]/(2a)
For the two roots ( distinguished by exp that's after + or - ) to be
equal,

sqrt(b^2 - 4ac) = 0
b^2 = 4ac

In our equation this condition becomes:
k^2 = 4x4x9
k = +/- 2x2x3 = +/- 12

Since k is given in the problem to be > 0,

k = 12

You can substitute k=12 in the given equation and verify it.

Answer 7

Kx 9