Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# More sum/product of roots questions?

One root of the equation x^2 + kx + 8 = 0 is twice the other. Find all possible values of k.

One root of the equation 4x^2 + kx + 3 = 0 is three times the other. Find all possible values of k.

Find a quadratic equation whose roots differ by 1 AND are reciprocals of each other. Find the roots of this equation.

The cubic equation 3x^3 - 5x - 3 = 0 has roots α, β, and γ. Write down the value of αβγ and show that α + β = 1/ αβ.

The quadratic equation 5x^2 - 3x + 2 = 0 has roots p and q. Find a quadratic equation with roots 1/p and 1/q.

Even if it's only a few (or just one), solving these would be a great help (my exam is tomorrow, I am so gonna fail if I can't figure these out).

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You get one question for your money. Here is the first.

x² + kx + 8 = 0, where one solution is twice the other

Let the solutions be r and 2r.

r + 2r = 3r = -k

(r)(2r) = 2r² = 8

-k = 3r

k² = 9r²

k² = 2r²(9/2)

k² = (8)(9/2)

k² = 36

k = ±6

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• x^2 + kx + 8 = 0, has values a = 1, b = k and c = 8.

Sum of roots = -b/a = -k/1

Product of roots = c/a = 8/1 = 8

Let r be one of the roots.

Then 2r is the other root.

Sum of the roots = 3r

Product of the roots = 2r^2

2r^2 = 8 means that r = 2 or -2.

This implies that 3r = 6 = k when r = 2.

This also implies 3r = -6 = k when r = -2

So k is either -6 or 6.

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