🧮 Algebra Quadratics And Discriminant
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Quadratic Formula · Easy · source: Original (AMC-style)
What is the discriminant of $x^2 - 4x + 3 = 0$?
A) $4$ B) $8$ C) $12$ D) $16$ E) $20$
Answer & Solution
Answer: A
The discriminant is $b^2 - 4ac = (-4)^2 - 4(1)(3) = 16 - 12 = 4$.
2.
Tags: Vertex Form · Easy · source: Original (AMC-style)
What is the vertex of $y = x^2 - 6x + 8$?
A) $(3, -1)$ B) $(3, 1)$ C) $(-3, -1)$ D) $(-3, 1)$ E) $(6, 8)$
Answer & Solution
Answer: A
Completing the square: $y = (x-3)^2 - 9 + 8 = (x-3)^2 - 1$. The vertex is $(3, -1)$.
3.
Tags: Discriminant · Easy · source: Original (AMC-style)
How many real solutions does $x^2 + 2x + 1 = 0$ have?
A) $0$ B) $1$ C) $2$ D) $3$ E) Infinitely many
Answer & Solution
Answer: B
The discriminant is $2^2 - 4(1)(1) = 0$, so there is exactly one real solution.
4.
Tags: Quadratic Formula · Easy · source: Original (AMC-style)
What are the solutions to $x^2 - 5x + 6 = 0$?
A) $x = 2, 3$ B) $x = -2, -3$ C) $x = 1, 6$ D) $x = -1, -6$ E) No real solutions
Answer & Solution
Answer: A
Factoring: $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.
5.
Tags: Discriminant Analysis · Medium · source: Original (AMC-style)
For what value of $k$ does $x^2 + kx + 9 = 0$ have exactly one real solution?
A) $-6$ B) $-3$ C) $0$ D) $3$ E) $6$
Answer & Solution
Answer: A
For exactly one real solution, the discriminant must be 0: $k^2 - 4(1)(9) = 0$, so $k^2 = 36$ and $k = \pm 6$.
6.
Tags: Vertex and Axis · Medium · source: Original (AMC-style)
What is the axis of symmetry of $y = 2x^2 - 8x + 5$?
A) $x = 1$ B) $x = 2$ C) $x = 3$ D) $x = 4$ E) $x = 5$
Answer & Solution
Answer: B
The axis of symmetry is $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = 2$.
7.
Tags: Quadratic Inequalities · Medium · source: Original (AMC-style)
For what values of $x$ is $x^2 - 4x + 3 > 0$?
A) $x < 1$ or $x > 3$ B) $1 < x < 3$ C) $x < -3$ or $x > -1$ D) $-3 < x < -1$ E) All real numbers
Answer & Solution
Answer: A
Factoring: $(x-1)(x-3) > 0$. The parabola opens upward, so it's positive when $x < 1$ or $x > 3$.
8.
Tags: Sum and Product · Medium · source: Original (AMC-style)
If the roots of $x^2 + px + q = 0$ are $r$ and $s$, and $r + s = 5$ and $rs = 6$, what is $p$?
A) $-5$ B) $-6$ C) $5$ D) $6$ E) $11$
Answer & Solution
Answer: A
By Vieta's formulas, $r + s = -p$ and $rs = q$. Since $r + s = 5$, we have $-p = 5$, so $p = -5$.
9.
Tags: Complex Discriminant · Hard · source: Original (AMC-style)
If the quadratic $x^2 + px + q$ has discriminant $D$ and $p^2 = 4q$, what is $D$?
A) $0$ B) $p^2$ C) $4q$ D) $p^2 - 4q$ E) Cannot be determined
Answer & Solution
Answer: A
The discriminant is $D = p^2 - 4q$. Since $p^2 = 4q$, we have $D = 4q - 4q = 0$.
10.
Tags: Quadratic with Parameters · Hard · source: Original (AMC-style)
If $x^2 + (k-1)x + k = 0$ has equal roots, what is $k$?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: A
For equal roots, the discriminant must be 0: $(k-1)^2 - 4(1)(k) = 0$. So $k^2 - 2k + 1 - 4k = 0$, giving $k^2 - 6k + 1 = 0$. Solving: $k = 3 \pm 2\sqrt{2}$.
11.
Tags: Advanced Discriminant · Hard · source: Original (AMC-style)
If $ax^2 + bx + c = 0$ has roots $\alpha$ and $\beta$ such that $\alpha^2 + \beta^2 = 10$ and $\alpha\beta = 3$, what is $b^2 - 4ac$?
A) $4$ B) $8$ C) $12$ D) $16$ E) $20$
Answer & Solution
Answer: A
We have $(\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2 = 10 + 2(3) = 16$. So $\alpha + \beta = \pm 4$. The discriminant is $b^2 - 4ac = a^2(\alpha + \beta)^2 - 4a^2\alpha\beta = a^2(16 - 12) = 4a^2$.
12.
Tags: Quadratic System · Hard · source: Original (AMC-style)
If $x^2 + y^2 = 25$ and $x^2 - y^2 = 7$, what is $x^2$?
A) $9$ B) $16$ C) $18$ D) $25$ E) $32$
Answer & Solution
Answer: B
Adding the equations: $2x^2 = 32$, so $x^2 = 16$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | A | B | A | A | B | A | A | A | A | A | B |