SOLUTION: Please help me with this question. I have to solve by whichever method is easier. ( factoring, completing the square or the quadratic formula) A) what is the discriminant of the

Question 896352: Please help me with this question. I have to solve by whichever method is easier. ( factoring, completing the square or the quadratic formula)
A) what is the discriminant of the equation 4x^2+8x+k=0
B) for what value of k will the equation have a double root?
C)for what values of k will the equation have two real roots?
D) for what values of k will the equation have imaginary roots?
E) name three values of k for which the given equation has rational roots

Found 2 solutions by richwmiller, MathTherapy:
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!

a=4 b=8 c=k
A) sqrt( 8^2-4*4*k ))

solve for k
B) 0 = sqrt( 8^2-4*4*k ) if d=0 there is one root (double root)
when k=4
C) 0< sqrt( 8^2-4*4*k ) if d is positive there are two real roots
when k<4
D) 0> sqrt( 8^2-4*4*k )if d is negative then there are imaginary roots
when k>4
E) k=1 k=2 k=3

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Start with the given expression.

Factor out the GCF .

Now let's try to factor the inner expression

---------------------------------------------------------------

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :

1

-1

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .

1*1 = 1
(-1)*(-1) = 1

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number Second Number Sum
1 1 1+1=2
-1 -1 -1+(-1)=-2

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

Condense the terms.

--------------------------------------------------

So then factors further to

===============================================================

Answer:

So completely factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .

1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number Second Number Sum
1 12 1+12=13
2 6 2+6=8
3 4 3+4=7
-1 -12 -1+(-12)=-13
-2 -6 -2+(-6)=-8
-3 -4 -3+(-4)=-7

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

===============================================================

Answer:

So factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Solved by pluggable solver: Quadratic Formula

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=4, b=8, and c=2

Square 8 to get 64

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

or

Now break up the fraction

or

Simplify

or

So the solutions are:

or

Solved by pluggable solver: Quadratic Formula

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=4, b=8, and c=1

Square 8 to get 64

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

or

Now break up the fraction

or

Simplify

or

So the solutions are:

or

Answer by MathTherapy(10552) (Show Source): You can put this solution on YOUR website!
Please help me with this question. I have to solve by whichever method is easier. ( factoring, completing the square or the quadratic formula)
A) what is the discriminant of the equation 4x^2+8x+k=0
B) for what value of k will the equation have a double root?
C)for what values of k will the equation have two real roots?
D) for what values of k will the equation have imaginary roots?
E) name three values of k for which the given equation has rational roots

Equation:
A) Discriminant: , or
B) 64 - 16k = 0___64 = 16k___k = 4. Double roots exist when k has a value of 4.
C) 64 � 16k ≥ 0____- 16k ≥ - 64____k ≤ , or k ≤ 4. Two real-roots exist when k has values ≤ 4.
D) 64 � 16k < 0____- 16k < - 64____k > , or k > 4. Imaginary roots exist when k has values > 4.
E) 64 � 16k ≥ 0, AND A PERFECT SQUARE. 64 � 16k ≥ 0____- 16k ≥ - 64____k ≤ , or k ≤ 4.
Rational roots exist when the k-values are ≤ 4, and DISCRIMINANT is a PERFECT SQUARE. Thus, three
values of k that are ≤ 4, and which produces RATIONAL ROOTS VALUES (PERFECT SQUARES) are:
You can do a check!!
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