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Difficulty: MediumQuadratic Equations and Polynomial Factoring

If the quadratic equation 3x2−kx+48=03x^2 - kx + 48 = 0 has equal real roots, where k>0k > 0, and the roots of the quadratic equation x2+kx−m=0x^2 + kx - m = 0 differ by 10, what is the value of mm?

Answer: -119

Answer

The value of mm is −119-119.
Setting the discriminant of 3x2−kx+48=03x^2 - kx + 48 = 0 equal to zero gives k2−4(3)(48)=0k^2 - 4(3)(48) = 0, yielding k=24k = 24 since k>0k > 0. Substituting k=24k = 24 into the second equation gives x2+24x−m=0x^2 + 24x - m = 0. According to Vieta's formulas, the sum of roots is r1+r2=−24r_1 + r_2 = -24 and the product of roots is r1r2=−mr_1 r_2 = -m. Using the algebraic identity (r1−r2)2=(r1+r2)2−4r1r2(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2 with the given root difference ∣r1−r2∣=10|r_1 - r_2| = 10 yields 102=(−24)2−4(−m)10^2 = (-24)^2 - 4(-m), which simplifies to 100=576+4m100 = 576 + 4m. Subtracting 576 from both sides gives 4m=−4764m = -476, so m=−119m = -119.

Step-by-Step Solution

1
Find kk using the discriminant condition for equal roots of 3x2−kx+48=03x^2 - kx + 48 = 0.
k=24k = 24
A quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 has equal real roots when its discriminant b2−4ac=0b^2 - 4ac = 0. Here, (−k)2−4(3)(48)=0(-k)^2 - 4(3)(48) = 0, so k2=576k^2 = 576. Since k>0k > 0, k=24k = 24.
2
Set up the algebraic relationship for the root difference of x2+24x−m=0x^2 + 24x - m = 0.
(r1−r2)2=100(r_1 - r_2)^2 = 100
Given that the difference between the roots r1r_1 and r2r_2 is 10, ∣r1−r2∣=10|r_1 - r_2| = 10, which implies (r1−r2)2=100(r_1 - r_2)^2 = 100.
3
Apply Vieta's formulas to express (r1−r2)2(r_1 - r_2)^2 in terms of mm and solve.
m=−119m = -119
By Vieta's formulas for x2+24x−m=0x^2 + 24x - m = 0, the sum of roots is r1+r2=−24r_1 + r_2 = -24 and the product of roots is r1r2=−mr_1 r_2 = -m. Using the identity (r1−r2)2=(r1+r2)2−4r1r2(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2, substitute the known values: 100=(−24)2−4(−m)=576+4m100 = (-24)^2 - 4(-m) = 576 + 4m. Solving 576+4m=100576 + 4m = 100 gives 4m=−4764m = -476, so m=−119m = -119.

Key Concept

Discriminant analysis and root difference identity via Vieta's formulas
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Topics

Quadratic Equations and Polynomial Factoring

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