SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots Questionbank

Practice SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

Prove that if the quadratic equation $x^2 + (p-5)x + 7 = 0$ has real and distinct roots, then $p < 5 - 2\sqrt{7}$ or $p > 5 + 2\sqrt{7}$ .

[7]

Question 2

SLPaper 1

The quadratic equation $(k+2)x^2 - 4x + (k-1) = 0$ , where $k eq -2$ and $k o \mathbb{R}$ , has real distinct roots.

Find the range of possible values for $k$ .

[5]

Question 3

SLPaper 1

Let $f(x)=x^{2}-6 x+7$ and $g(x)=f(x)+k$ , where $k \in \mathbb{R}$ .

Write down the expression for $g(x)$ .

[1]

Find the values of $k$ such that $g(x)=0$ has no real roots.

[3]

Using the value $k=2$ , determine the minimum value of $g(x)$ .

[2]

Question 4

SLPaper 1

Consider the quadratic equation $3x^2 + 2x - 1 = 0$ .

Solve the equation.

[3]

Question 5

SLPaper 1

The graph of a quadratic function $f$ has $x$ -intercepts at $x=-6$ and $x=2$ . The vertex of the graph lies on the line $y=-x+1$ .

Find an expression for $f(x)$ .

[5]

Determine the range of $f$ .

[2]

The domain of $f$ is restricted to $x \le -2$ . Find an expression for $f^{-1}(x)$ .

[5]

Describe a sequence of transformations that maps the graph of $y=x^2$ to the graph of $f$ .

[3]

Question 6

SLPaper 1

The quadratic function $f$ has $x$ -intercepts $x=1$ and $x=9$ , and its vertex lies on the line $y=3$ .

Find $f(x)$ .

[5]

Determine the range of $f(x)$ .

[2]

Find $f^{-1}(x)$ given the domain $x \le 5$ .

[5]

Describe the transformations to get $f(x)$ from $y=x^2$ .

[3]

Question 7

SLPaper 2

For a real parameter $p$ , consider the quadratic $q_p(x)=x^2-(p+2)x+(2p-5).$

Find all real values of $p$ for which $q_p$ has two distinct real roots and both roots are greater than $0$ .

[6]

Let $p=4$ so that $q(x)=x^2-6x+3$ , and let $g(x)=\ln(x+2)+1$ (domain $x>-2$ ).

Find the exact roots of $q(x)=0$ .

[3]

Solve $q(x)=g(x)$ for $x\ge0$ to three decimal places. Justify the number of solutions on $x\ge0$ .

[6]

Solve $q(x)\ge g(x)$ for $x\ge0$ (endpoints to 3 d.p.).

[3]

Question 8

SLPaper 1

A quadratic has roots $x=-4$ and $x=8$ ; its vertex lies on $y=x-6$ .

Find $f(x)$ .

[5]

Determine the range of $f(x)$ .

[2]

Find $f^{-1}(x)$ with appropriate restriction.

[5]

Describe the transformations to obtain $f(x)$ from $y=x^2$ .

[3]

Question 9

SLPaper 1

The quadratic has $x$ -intercepts $x=2$ and $x=10$ , and its vertex lies on the line $y=-x+9$ .

Find $f(x)$ .

[5]

Determine the range of $f(x)$ .

[2]

The function $f$ is defined for $x \le 6$ . Find an expression for $f^{-1}(x)$ .

[5]

Describe transformations to obtain $f(x)$ from $y=x^2$ .

[3]

Question 10

HLPaper 2

Consider the quadratic equation $mx^2 - (m + 3)x + 2m + 9 = 0$ , where $m \in \mathbb{R}$ .

Write down an expression for the product of the roots, in terms of $m$ .

[1]

Hence or otherwise, determine the values of $m$ such that the equation has one positive and one negative real root.

[3]

Find for what values there is one distinct real root.

[3]