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Zorluk: ZorQuadratic Equations and Polynomial Factoring

If xx is a real number satisfying the equation (x2−2x)2−(x2−2x)−12=0(x^2 - 2x)^2 - (x^2 - 2x) - 12 = 0, what is the product of all distinct real roots of the equation?

  1. A
    −12-12
  2. −4-4Cevap
  3. C
    −3-3
  4. D
    22
  5. E
    1212

Cevap

The product of all distinct real roots of the equation is −4-4.
Substituting u=x2−2xu = x^2 - 2x simplifies the given equation to u2−u−12=0u^2 - u - 12 = 0, which factors as (u−4)(u+3)=0(u - 4)(u + 3) = 0. Setting x2−2x=4x^2 - 2x = 4 yields x2−2x−4=0x^2 - 2x - 4 = 0, which has two real roots (since the discriminant is 20>020 > 0) with product −4-4. Setting x2−2x=−3x^2 - 2x = -3 yields x2−2x+3=0x^2 - 2x + 3 = 0, which has a negative discriminant (−8-8) and therefore no real solutions. Thus, the product of all distinct real roots is −4-4.

Adım Adım Çözüm

1
Use algebraic substitution to simplify the disguised quadratic equation.
Let u=x2−2xu = x^2 - 2x. Substituting uu into (x2−2x)2−(x2−2x)−12=0(x^2 - 2x)^2 - (x^2 - 2x) - 12 = 0 gives u2−u−12=0u^2 - u - 12 = 0.
Recognizing the repeated expression x2−2xx^2 - 2x reduces a 4th-degree polynomial into a standard 2nd-degree quadratic equation in terms of uu.
2
Solve for uu by factoring.
u2−u−12=(u−4)(u+3)=0u^2 - u - 12 = (u - 4)(u + 3) = 0, so u=4u = 4 or u=−3u = -3.
Factoring provides the possible numerical values for the expression x2−2xx^2 - 2x.
3
Analyze each case for xx and check for real roots using the discriminant.
Case 1: x2−2x=4  ⟹  x2−2x−4=0x^2 - 2x = 4 \implies x^2 - 2x - 4 = 0. Discriminant D1=(−2)2−4(1)(−4)=20>0D_1 = (-2)^2 - 4(1)(-4) = 20 > 0. Since D1>0D_1 > 0, this quadratic has 2 distinct real roots. By Vieta's formula, the product of these two real roots is ca=−41=−4\frac{c}{a} = \frac{-4}{1} = -4.
Case 2: x2−2x=−3  ⟹  x2−2x+3=0x^2 - 2x = -3 \implies x^2 - 2x + 3 = 0. Discriminant D2=(−2)2−4(1)(3)=−8<0D_2 = (-2)^2 - 4(1)(3) = -8 < 0. Since D2<0D_2 < 0, this quadratic has no real roots.
Only roots with a non-negative discriminant belong to the real number domain and contribute to the product of real roots.
4
Determine the product of all distinct real roots.
The total product of all distinct real roots is −4-4.
Since Case 1 produces the only real roots of the original equation, their product −4-4 is the complete product of all real solutions.

Anahtar Kavram

Disguised Quadratic Equations and Discriminant Real Root Filtering
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Quadratic Equations and Polynomial Factoring

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