# Given the equation, find the values of k such that the equation has...?

Given the equation x^2 + kx - k + 8 = 0, find the values of k such that the equation has:

1. equal roots

2. real roots

3. no real roots

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• Anonymous
9 years ago

1.For an equation to have equal roots, the discriminant (b^2-4ac) must be 0

a=1 b=k c=(-k+8)

k^2-4(1)(-k+8)=0

k^2+4k-32=0

(k+8)(k-4)=0

k=-8,4

2.For real roots, it must be > 0

k^2+4k-32>0

(k+8)(k-4)>O

I would use plus/minus lines here, but i cannot type one, so ill skip this step, sorry

in interval notation, the answer would be:

(-infinite,-8)u(4,infinite), meaning any number except -8 through 4 will work

3.For no real roots, it must be < 0

(k+8)(k-4)<0

Again, plus/minus lines, sorry

(-8,4), meaning anything from -8 to 4 will work

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• Use the discriminant: b^2 - 4ac. in case you have an equation in one in all those ax^2 + bx + c = 0, it could have 2 unique recommendations if the discriminant is larger than 0, one answer if the discriminant is comparable to 0, and no recommendations if it is below 0. So regarding x^2 + kx + seventy 9=0, we've a=a million, b=ok, and c=seventy 9. Plug this into the discriminant and you get ok^2 - 4(a million)(seventy 9), or ok^2 - 316. So for the unique equation to in basic terms have one root, this ought to equivalent 0. it rather is actual while ok = sqrt(316) or -sqrt(316). the explanation the discriminant works will become sparkling in case you look on the quadratic formulation. The discriminant is the section got here upon below the novel. in case you could take the sq. root, then the "plus or minus" in the quadratic formulation provides 2 solutions. If the discriminant is 0 then the sq. root of it rather is 0, and the "plus or minus" 0 in the quadratic formulation in basic terms provides the same answer two times. If the discriminant is below 0, then you definately have not got any answer, when you consider which you could no longer take the sq. root of a adverse extensive kind.

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• The number of roots of a quadratic is determined by the discriminant b^2 - 4ac where

a = coefficient of the x^2 term (1 for you)

b = coefficient of the x term (k for you)

c = constant term (-k + 8 for you).

So the discriminant is b^2 - 4ac = k^2 - 4*1*(-k+8)

There is only one root, or two equal roots, when this is equal to 0. Set it equal to 0 and solve for k.

There are two roots when this is greater than 0. Solve that inequality.

There are no real roots when this is less than 0. Solve that inequality.

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