Find all integers ( k ) such that the trinomial ( x^2 - kx + 12 ) can be factored over the integers.

Question

Accepted Answer

Step:1 ：The trinomial can be factored over the integers if and only if the discriminant ( Delta = b^2 - 4ac ) is a perfect square, where ( a, b, c ) are the coefficients of the quadratic equation.Step:2 ：In this case, ( a = 1 ), ( b = -k ), and ( c = 12 ). Thus, ( Delta = (-k)^2 - 4cdot1cdot12 = k^2 - 48 ).Step:3 ：For ( Delta ) to be a perfect square, ( k^2 - 48 ) must be a perfect square. This happens when ( k ) is an integer such that ( k^2 ) is 48 more than a perfect square.Step:4 ：By checking integers, we can find that the integers that satisfy this condition are ( k = -8, -6, 6, 8 ), since ( (-8)^2 = 64 ), ( (-6)^2 = 36 ), ( 6^2 = 36 ), and ( 8^2 = 64 ), all of which are 48 more than a perfect square.

Answer

Step: 1 ：The trinomial can be factored over the integers if and only if the discriminant ( Delta = b^2 - 4ac ) is a perfect square, where ( a, b, c ) are the coefficients of the quadratic equation.Step: 2 ：In this case, ( a = 1 ), ( b = -k ), and ( c = 12 ). Thus, ( Delta = (-k)^2 - 4cdot1cdot12 = k^2 - 48 ).Step: 3 ：For ( Delta ) to be a perfect square, ( k^2 - 48 ) must be a perfect square. This happens when ( k ) is an integer such that ( k^2 ) is 48 more than a perfect square.Step: 4 ：By checking integers, we can find that the integers that satisfy this condition are ( k = -8, -6, 6, 8 ), since ( (-8)^2 = 64 ), ( (-6)^2 = 36 ), ( 6^2 = 36 ), and ( 8^2 = 64 ), all of which are 48 more than a perfect square.

Find all integers $k$ such that the trinomial $x^{2} - k x + 12$ can be factored over the integers.

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Popular Problems

Find all integers k such that the trinomial x2−kx+12 can be factored over the integers.

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Popular Problems

Find all integers $k$ such that the trinomial $x^{2} - k x + 12$ can be factored over the integers.