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SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots - IB Questionbank | RevisionDojo

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 $(k+2) x^{2}-4 x+(k-1)=0$ , where $k \in \mathbb{R}$ , 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]

For the smallest value of $k$ found in part 2, determine the minimum value of $g(x)$ .

[2]

Question 4

SLPaper 1

A quadratic has $x$ -intercepts $x=-6$ and $x=2$ , and its vertex lies on $y=-x+1$ .

Find $f(x)$ .

[5]

Determine the range of $f(x)$ .

[2]

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

[5]

Transformations from $y=x^2$ .

[3]

Question 5

SLPaper 1

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

Find $f(x)$ .

[5]

Determine the range of $f(x)$ .

[2]

Find $f^{-1}(x)$ with a suitable domain restriction.

[5]

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

[3]

Question 6

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]

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 7

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 8

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]

Find $f^{-1}(x)$ with a suitable domain restriction.

[5]

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

[3]

Question 9

SLPaper 1

A quadratic has $x$ -intercepts $x=0$ and $x=8$ , and its vertex lies on the line $y=2x-4$ .

Find $f(x)$ .

[5]

Determine the range of $f$ .

[2]

Find $f^{-1}(x)$ by reflecting in $y=x$ , with a suitable domain restriction.

[5]

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

[3]

Question 10

SLPaper 1

Solve the quadratic inequality $x^{2}+2 x-3 \leq 0$ .

[5]

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