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Zorluk: ZorQuadratic Equations and Factoring

Let f(x)=x2+bx+cf(x) = x^2 + bx + c, where bb and cc are integers. The equation f(x)=0f(x) = 0 has two distinct real roots, α\alpha and β\beta. If α+β=αβ\alpha + \beta = \alpha\beta and c>0c > 0, which of the following statements must be true? Indicate all such statements.

  1. The constant cc is strictly greater than 4.Cevap
  2. Both roots α\alpha and β\beta are positive.Cevap
  3. The vertex of the parabola y=f(x)y = f(x) lies in Quadrant IV of the xy-plane.Cevap
  4. D
    The coefficient bb is positive.
  5. E
    The difference between the two roots, ∣α−β∣|\alpha - \beta|, can equal 2.

Cevap

The statements asserting that the constant cc is strictly greater than 4, both roots are positive, and the vertex lies in Quadrant IV must all be true.
By Vieta's formulas, α+β=−b\alpha + \beta = -b and αβ=c\alpha\beta = c. Equating sum and product gives b=−cb = -c. The discriminant Δ=c2−4c>0\Delta = c^2 - 4c > 0 requires c>4c > 4 because c>0c > 0. Since sum and product of the roots equal c>0c > 0, both roots are positive. The vertex coordinates (c/2,−c(c−4)/4)(c/2, -c(c-4)/4) have a positive x-value and a negative y-value, placing the vertex in Quadrant IV.

Adım Adım Çözüm

1
Apply Vieta's formulas and establish the relationship between coefficients
α+β=−b\alpha + \beta = -b and αβ=c\alpha\beta = c. Setting them equal gives −b=c  ⟹  b=−c-b = c \implies b = -c.
The stem specifies that the sum of the roots equals their product.
2
Analyze the discriminant for distinct real roots
Δ=b2−4c=(−c)2−4c=c(c−4)>0\Delta = b^2 - 4c = (-c)^2 - 4c = c(c - 4) > 0. Since c>0c > 0, c−4>0  ⟹  c>4c - 4 > 0 \implies c > 4.
Two distinct real roots require a strictly positive discriminant.
3
Determine the signs of the roots
α+β=c>0\alpha + \beta = c > 0 and αβ=c>0\alpha\beta = c > 0, implying α>0\alpha > 0 and β>0\beta > 0.
If the sum and product of two real numbers are both positive, both numbers must be positive.
4
Find the location of the parabola's vertex
xv=c2>0x_v = \frac{c}{2} > 0 and yv=−c(c−4)4<0y_v = -\frac{c(c-4)}{4} < 0, placing the vertex in Quadrant IV.
A point with a positive x-coordinate and negative y-coordinate resides in the fourth quadrant.
5
Evaluate the remaining options regarding bb and ∣α−β∣|\alpha - \beta|
b=−c<−4b = -c < -4 (negative), and ∣α−β∣=2  ⟹  c=2+22|\alpha - \beta| = 2 \implies c = 2 + 2\sqrt{2}, which is not an integer.
These evaluations disprove the statements that bb is positive and that ∣α−β∣|\alpha - \beta| can equal 2.

Anahtar Kavram

Quadratic Root Properties and Vieta's Formulas
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Quadratic Equations and Factoring

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