Quadratic Equation Level - 1
Q1–10 of 60Find the maximum value of the expression 1 / (x² + 5x + 10)
Paliwanag: Para makuha ang maximum value ng buong fraction, kailangang maging minimum value ang denominator na $x^2 + 5x + 10$.
Solusyon:
Para sa quadratic $ax^2 + bx + c$, ang minimum value nito ay $\frac{4ac - b^2}{4a}$.
Sagot: (c) 4/15
Find the maximum value of the expression (x² + 8x + 20)
Paliwanag: Dahil ang coefficient ng $x^2$ ay positive ($a = 1 > 0$), ang graph ng parabola ay nakabukas pataas. Ibig sabihin, tumataas ito patungong $+\infty$ kaya walang maximum value (mayroon lamang minimum value).
Sagot: (d) None of these
Find the minimum value of the expression (p + 1/p); p > 0
Paliwanag: Gamitin ang AM-GM Inequality ($\text{Arithmetic Mean} \ge \text{Geometric Mean}$) para sa mga positibong numero:
Sagot: (c) 2
If the product of roots of the equation x² – 3(2a + 4)x + a² + 18a + 81 = 0 is unity, then a can take the values as
Paliwanag: Ang product of roots ng $x^2 + Bx + C = 0$ ay ang constant term $C$.
Sagot: (c) -10, -8
For the equation 2(a+3) = 4(a+2) – 48, the value of a will be
Paliwanag: Isulat ang $4^{a+2}$ bilang $(2^2)^{a+2} = 2^{2a+4}$.
Hayaan ang $y = 2^a$:
Dahil ang $2^a$ ay dapat positibo: $y = 2 \implies 2^a = 2^1 \implies a = 1$.
Sagot: (d) 1
The expression a² + ab + b² is ______ for a < 0, b < 0
Paliwanag: I-complete ang square:
Dahil ang parehong terms ay squares ng tunay na numero at $b < 0$, ang resulta ay laging positibo ($>0$).
Sagot: (c) > 0
If the roots of equation x² + bx + c = 0 differ by 2, then which of the following is true?
aliwanag: Ang pagkakaiba ng mga roots ay $\vert{}\alpha - \beta\vert{} = \frac{\sqrt{b^2 - 4c}}{1} = 2$.
Sagot: (d) $b^2 = 4(c + 1)$
If f(x) = (x + 2) and g(x) = (4x + 5), and h(x) is defined as h(x) = f(x) g(x), then sum of roots of h(x) will be
Paliwanag: Ang mga roots ay $x = -2$ at $x = -5/4$.
Sagot: (c) -13/4
If equation x² + bx + 12 = 0 gives 2 as its one of the roots and x² + bx + q = 0 gives equal roots then the value of b is
Paliwanag: I-substitute ang $x = 2$ sa unang equation:
$$2^2 + b(2) + 12 = 0 \implies 2b = -16 \implies b = -8$$Sagot: (b) -8
If the roots of the equation (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal then which of the following is true?
Paliwanag: Kapag equal ang roots, Discriminant $D = 0$:
Sagot: (b) ad = bc