Quadratic Function practice questions

Question 1

What is the sum of the solutions to $x^2 - 5x + 6 = 0$?

Accepted Answer

Factoring gives $(x-2)(x-3)=0$, so the roots are $2$ and $3$. Their sum is $5$. (Equivalently, $-b/a = 5/1$.)

Answer

$-5$

Answer

$-1$

Answer

$1$

Answer

$5$

Question 2

The discriminant of $2x^2 - 3x - 5 = 0$ equals:

Accepted Answer

$\Delta = (-3)^2 - 4 \cdot 2 \cdot (-5) = 9 + 40 = 49$.

Answer

$49$

Answer

$9$

Answer

$-31$

Answer

$29$

Question 3

The roots of $x^2 - 5x + 6 = 0$ are:

Accepted Answer

$(x - 2)(x - 3) = 0 \Rightarrow x \in \{2, 3\}$.

Answer

$1$ and $6$

Answer

$-2$ and $-3$

Answer

$2$ and $3$

Answer

$-1$ and $-6$

Question 4

The vertex (extremum point) of $f(x) = x^2 - 4x + 3$ has $x$-coordinate:

Accepted Answer

$x_V = -\dfrac{-4}{2 \cdot 1} = 2$. (And $f(2) = 4 - 8 + 3 = -1$ is the minimum.)

Answer

$-4$

Answer

$-2$

Answer

$2$

Answer

$4$

Question 5

Solve by factorising: $x^2 + 7x + 12 = 0$.

Accepted Answer

Look for two numbers with product $12$ and sum $7$: these are $3$ and $4$.
$$x^2 + 7x + 12 = (x+3)(x+4)$$
$$(x+3)(x+4) = 0$$
$$x = -3 	ext{ or } x = -4$$

Answer

$x = -3 	ext{ or } x = -4$

Answer

$x = 3 	ext{ or } x = 4$

Answer

$x = -2 	ext{ or } x = -6$

Answer

$x = -1 	ext{ or } x = -12$

Question 6

The quadratic equation $x^2 - 7x + 12 = 0$ has two solutions. What is the sum of its solutions?

Accepted Answer

Factor the quadratic:
$$x^2 - 7x + 12 = (x - 3)(x - 4) = 0$$

So the solutions are:
$$x = 3 \quad 	ext{or} \quad x = 4$$

Their sum is:
$$3 + 4 = 7$$

This matches Vieta's formula, $-b/a = -(-7)/1 = 7$.

Answer

$-7$

Answer

$7$

Answer

$12$

Answer

$1$

Question 7

For which values of $m \in \mathbb{R}$ does $x^2 + (m - 2)x + 1 = 0$ have a double root?

Accepted Answer

$\Delta = (m - 2)^2 - 4 = 0 \Rightarrow (m - 2)^2 = 4 \Rightarrow m - 2 = \pm 2 \Rightarrow m \in \{0, 4\}$.

Answer

Only $m = 0$

Answer

Only $m = 4$

Answer

$m \in \{-2, 2\}$

Answer

$m \in \{0, 4\}$

Question 8

For $f: \mathbb{R} 	o \mathbb{R}$, $f(x) = 3x^2 - 7x + 2$, the value of $f(2025) \cdot f(2)$ is:

Accepted Answer

$f(2) = 12 - 14 + 2 = 0$. So $f(2025) \cdot f(2) = f(2025) \cdot 0 = 0$, regardless of $f(2025)$.

Answer

$0$

Answer

$1$

Answer

$f(2025)$

Answer

A non-zero value depending on $2025$

Question 9

The solution set of $x^2 - 4 < 0$ in $\mathbb{R}$ is:

Accepted Answer

$(x - 2)(x + 2) < 0 \Leftrightarrow x \in (-2, 2)$.

Answer

$(-\infty, -2) \cup (2, +\infty)$

Answer

$\{-2, 2\}$

Answer

$(-2, 2)$

Answer

$[-2, 2]$

Question 10

The minimum value of $f: \mathbb{R} 	o \mathbb{R}$, $f(x) = x^2 - 6x + 8$ is:

Accepted Answer

$\Delta = 36 - 32 = 4$, $a = 1$, so $f_{\min} = -\dfrac{4}{4} = -1$ (attained at $x = 3$).

Answer

$-1$

Answer

$0$

Answer

$1$

Answer

$8$

Quadratic Function

Problems & worked solutions

Problem 1 — Quadratic Equations

Problem 2 — Quadratic Function

Problem 3 — Quadratic Function

Problem 4 — Quadratic Function

Problem 5 — Quadratic Equations

Problem 6 — Quadratic Equations

Problem 7 — Quadratic Function

Problem 8 — Quadratic Function

Problem 9 — Quadratic Function

Problem 10 — Quadratic Function

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