Zadanie 2a
Write f(x) in the form (x + p)2 + q, with integers p,q.
Quadratic solutions and discriminants
Solve quadratic equations and inequalities by factorization, completing the square or the quadratic formula; use the discriminant to classify roots and constrain parameters.
Tę umiejętność sprawdza 56 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 56 Otwórz w wyszukiwarce z filtrami
Zadanie 2b
State the vertex coordinates of y=f(x).
Zadanie 2c
Hence find the minimum of .
Zadanie 3
A differentiable function has derivative f′(x) = 2x2 + x − 4 for all real x. Find where f is decreasing.
Zadanie 6b
A and B are independent. Find all possible x.
Zadanie 7e
Hence solve f(x)=f⁻¹(x).
Zadanie 7a
Find (i) the x-coordinates of the x-intercepts; (ii) the vertex coordinates.
- Materiał graficzny
Zadanie 7b
Write f in the form −2(x − h)2 + k by stating (i) h; (ii) k.
Zadanie 7c
The line y=mx+12 meets this parabola at x=−3 and x=3. Find m.
Zadanie 7d
For the equation −2x2 + 4x + 30 − p = 0, determine the range of p giving (i) two distinct real roots; (ii) two distinct positive real roots.
Zadanie 8b
The graph has horizontal tangents at x=3 and at a distinct x=k. (i) Show that c=9. (ii) Find k. (iii) Classify the stationary point at x=k, with justification.
Zadanie 9d
Given h′′(x) = e3x(9x2 − 6x + 2)/x3, prove that the curve has no inflection point.
Zadanie 9c
(i) Show that 69r2 − 98r + 29 = 0. (ii) Find r. (iii) Hence find a₁.
Zadanie 5a
Find the allowable values of k.
Zadanie 5b
Find the allowable values of k if the two solutions also have the same sign.
Zadanie 5a
Show that their intersection coordinates satisfy 2x2 + (3 − m)x + 3 = 0.
Zadanie 5b
Hence determine every slope m for which the line is tangent to the parabola.
Zadanie 7a
Prove that k = 1/4.
Zadanie 8c
Solve f(x) ≤ 24.
Zadanie 8d
(i) Write an expression for (f∘g)(x). (ii) Solve (f∘g)(x) ≤ 24.
Zadanie 8d
Rods are supplied in boxes of 50. Find the maximum number of tiers that can be built using 12 boxes.
Zadanie 8c
The already-investigated geometric case is k = −6. A second value of k also makes the sequence geometric. (i) Show that k2 + 7k + 6 = 0. (ii) Find the first three terms for the second value. (iii) State the sum S2m of its first 2m terms.
Zadanie 9a
(i) Solve 24 + 2x − x2 = 0. (ii) Hence solve 24 + 2x − x2 > 0.
Zadanie 9c
Find the exact x-coordinates where f(x) = 0.
Zadanie 4a
(i) State c. (ii) Find a.
- Materiał graficzny
Zadanie 6a
Show that p=−4.
Zadanie 6b
Find q.
Zadanie 8a
Show that the greatest possible k is 6.
Zadanie 8b
Take k=6. Find (i) the equation of the axis of symmetry; (ii) the coordinates of the minimum.
Zadanie 9c
Use the identity to prove x²−4x+1=0.
Zadanie 9d
Hence find tan15° exactly.
Zadanie 3a
Solve 2m2 + 7m − 4 = 0.
Zadanie 3b
Hence or otherwise, solve 2 ⋅ 16x + 7 ⋅ 4x − 4 = 0 for real x.
Zadanie 5b
(i) Show that 16k2 − 40k + 9 = 0. (ii) The roots of this equation are 1/4 and 9/4. State the valid value of k and justify your choice.
Zadanie 6b
(i) For m > 0, state the number of real solutions of f(x) = g(x). (ii) Find m for which there is exactly one real solution. (iii) Find all m for which there are two distinct solutions and both are nonnegative.
Zadanie 8c
Show that p=8/3.
Zadanie 8d
Find b and c.
Zadanie 8e
Find the y-coordinate of the vertex of g.
Zadanie 8f
Find the number of solutions of f(x)=g(x).
Zadanie 8a
(i) Show that 2x2 − 8x + 11 > 0 for every real x. (ii) Hence determine where f is increasing.
Zadanie 6
The function h(x) = log2(5x2 − rx + 2r − 3) must be real-valued for every real x. Find all possible r.
Zadanie 6
The expression h(x) = log3(3x2 − rx + r − 2) must define a real-valued function for every real x. Find all possible real r.
Zadanie 5
Find all real k for which e2x + ln k = 4ex has at least one real solution x.
Zadanie 7e
Find the distance PQ exactly.
- Materiał graficzny
Zadanie 7b
Find the x-coordinates where the two graphs meet.
Zadanie 1c
Find B. You may use the normal equation y=x/2−27/2.
Zadanie 1c
Find B. You may use the normal equation y=x/2−27/2.
Zadanie 8b
(i) Show that intersection abscissas satisfy x2 − 3dx + 6d = 0. (ii) Hence show that 9d2 − 24d > 0. (iii) Find the possible values of d.
Zadanie 8c
For d=4, find q−p in simplest surd form. Intersection abscissas satisfy x2 − 3dx + 6d = 0.
Zadanie 8b
(i) Show that intersection abscissas satisfy x2 − 3dx + 6d = 0. (ii) Hence show that 9d2 − 24d > 0. (iii) Find the possible values of d.
Zadanie 8c
For d=4, find q−p in simplest surd form. Intersection abscissas satisfy x2 − 3dx + 6d = 0.
Zadanie 6b
The third term is the arithmetic mean of the second and fourth terms. Find all possible x.
Zadanie 7b
(i) State the discriminant of the quadratic f′. (ii) Hence or otherwise find p.
Zadanie 7d
Find d for which g has a strictly positive derivative at every real x.
Zadanie 8a
Find the coordinates of B.
Zadanie 7c
Without the restrictions g(−4)=g(2)=12, consider all real t and all real p≠0. Prove that the graphs of g and j(x) = −2x + 5p have two distinct intersections.