quadratic equations and functions

Sub-topic 1 2025 · National One-Eighth Free

Solving $(x-2)^{x-1}=1$, $(x+2)^{x-2}=1$ and $(2x-1)^{x-3}=1$

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Sub-topic 2 2024 · National Preliminary Premium

Least and greatest values of $x^2+6x-7$ on $2\leq x\leq5$ (vertex outside the interval)

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Sub-topic 3 2025 · National One-Eighth Premium

Factorised equation of the parabola with $x$-intercepts $1$, $3$ and $y$-intercept $9$

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Sub-topic 4 2025 · National One-Eighth Premium

Factorised equation of a parabola from its intercepts, e.g. $x=\pm2$ and $y=-8$

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Sub-topic 5 2025 · National One-Eighth Premium

Riddle: the quadratic equation whose roots have sum $3\times5$ and product $3+5$

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Sub-topic 6 2025 · National One-Eighth Premium

Quadratic equation with integer coefficients having the root $3+\sqrt{7}$

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Sub-topic 7 2025 · Regional Championship Premium

Quadratic equation with integer coefficients having the root $2-\sqrt{7}$

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Sub-topic 8 2025 · National One-Eighth Premium

Factorised equation with $x$-intercepts $-3$, $5$ and $y$-intercept $30$ (speed race)

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Sub-topic 9 2024 · National One-Eighth Premium

Is $\alpha^2+\beta^2=b^2+4c$ for the roots of $x^2+bx+c=0$? (true/false)

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Sub-topic 10 2024 · National Preliminary Premium

Least and greatest values of $x^2+6x+12$ on $-4\leq x\leq2$ (vertex inside the interval)

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Sub-topic 11 2024 · National Preliminary Premium

$(\alpha-\beta)^2$ for the roots of $x^2-3x+5=0$

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Sub-topic 12 2024 · National Preliminary Premium

Constants $a$, $b$, $c$ in $a-b(x+c)^2$ equal to a given quadratic

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Sub-topic 13 2026 · National One-Eighth Premium

Constants in $a(x-3)^2+b(x-3)+c=2x^2-5x-12$

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Sub-topic 14 2026 · National One-Eighth Premium

Factor theorem: is a quadratic a factor of a cubic polynomial? (true/false)

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Sub-topic 15 2026 · National Preliminary Premium

Solving $x+y=8$ and $y=x^2-12$

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Sub-topic 16 2025 · National Preliminary Premium

Constants in $3x^2+12x+20=a(x+b)^2+c$

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Sub-topic 17 2026 · National Semifinal Premium

Compound inequality $x+3\lt x^2+1\lt10$

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Sub-topic 18 2026 · National Semifinal Premium

Solving $|x^2-2|=x$

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Sub-topic 19 2025 · National Semifinal Premium

Constants $p$ and $q$ when $x^2-2x+3$ divides $3x^3+px^2+qx+15$

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Sub-topic 20 2025 · National Semifinal Premium

Maximum product of two numbers with sum $16$

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Sub-topic 21 2024 · National Semifinal Premium

Compound inequality $16\lt x^2\lt36$

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Sub-topic 22 2025 · National Semifinal Premium

Solving $|x^2-30|=x$

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Sub-topic 23 2026 · National Quarterfinal Premium

$x^2+xy+y^2$ from $x^3-y^3=56$ and $x-y=4$

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Sub-topic 24 2026 · National Quarterfinal Premium

Vertex of $y=3x^2-6x+7$

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Sub-topic 25 2026 · National Quarterfinal Premium

Compound inequalities such as $1\lt x^2-3\lt33$

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Sub-topic 26 2026 · National Quarterfinal Premium

Solving absolute value equations such as $|x^2-3x|=4$

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Sub-topic 27 2026 · National One-Eighth Premium

Constants $p$ and $q$ when $px^2+qx-8\lt0$ has solution set $-2\lt x\lt4$

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Sub-topic 28 2026 · National One-Eighth Premium

Solving $(x-1)(x+2)\gt4$

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Sub-topic 29 2026 · National One-Eighth Premium

Factorised equation with intercepts $(-2,0)$, $(3,0)$ and $(0,-12)$

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Sub-topic 30 2018 · National One-Eighth Premium

Values of $k$ for which $3x^2+2kx+2k=0$ has no real roots

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Sub-topic 31 2026 · National Preliminary Premium

Values of $k$ for which $x^2+2kx+16$ is positive for all real $x$

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Sub-topic 32 2026 · National Preliminary Premium

Greatest value of $-6+12x-x^2$ and the corresponding $x$

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Sub-topic 33 2026 · National Preliminary Premium

Maximum value of $25+10x-x^2$ and the corresponding $x$

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Sub-topic 34 2026 · National Preliminary Premium

Solving $(x+4)(x-3)\lt8$

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Sub-topic 35 2024 · National One-Eighth Premium

Constants in $10-12x-2x^2=a-b(x+c)^2$

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Sub-topic 36 2026 · National One-Eighth Premium

Solving $(x-5)(x+2)\lt18$

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Sub-topic 37 2024 · Regional Championship Premium

Curve with a stationary point on the $x$-axis at $a(1,0)$ and $y$-intercept $b(0,6)$

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Sub-topic 38 2026 · National Preliminary Premium

Nature of the roots of $x^2+(k-3)x+k=0$ for varying $k$

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Sub-topic 39 2024 · National Preliminary Premium

$(\alpha+\beta)^2$ for the roots of $x^2-3x+5=0$

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Sub-topic 40 2024 · National Preliminary Premium

Sum, product and nature of the roots of a quadratic (true/false)

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Sub-topic 41 2012 · National Preliminary Premium

Inverse variation: $y\propto\dfrac{1}{x^2}$ with $y=1$ when $x=5$

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Sub-topic 42 2012 · National Preliminary Premium

Direct variation: $y\propto x^3$ with $y=1$ when $x=-2$

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Sub-topic 43 2013 · National Preliminary Premium

Minimum sum of two positive numbers with product $100$

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Sub-topic 44 2013 · National Preliminary Premium

Maximum product of two numbers with sum $10$

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Sub-topic 45 2013 · National Preliminary Premium

Symmetric functions of the roots of $x^2+bx+c=0$ and a derived quadratic

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Sub-topic 46 2016 · National Preliminary Premium

Riddle: a three-digit number built from the roots of $x^2-3x+5=0$

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Sub-topic 47 2017 · National Quarterfinal Premium

Completing the square: $4x^2-16x+21=a(x+b)^2+c$

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Sub-topic 48 2017 · National Quarterfinal Premium

Completing the square: $3x^2+18x+10=a(x+b)^2+c$

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Sub-topic 49 2017 · National Quarterfinal Premium

Completing the square: $2x^2-20x+37=a(x+b)^2+c$

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Sub-topic 50 2017 · National Quarterfinal Premium

Does $x^2+5x-4=0$ have real roots? (true/false)

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Sub-topic 51 2017 · National Quarterfinal Premium

Does $x^2+5x+4=0$ have rational roots? (true/false)

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Sub-topic 52 2017 · National Quarterfinal Premium

Does $x^2+x+2=0$ have no real roots? (true/false)

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Sub-topic 53 2018 · National Preliminary Premium

Maximum or minimum point of $y=3+2x-x^2$

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Sub-topic 54 2018 · National Preliminary Premium

Maximum or minimum point of $y=2x^2-4x+5$

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Sub-topic 55 2018 · National Preliminary Premium

Maximum or minimum point of $y=x^2+4x+5$

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Sub-topic 56 2020 · National Preliminary Premium

Vertex of $y=-3x^2+18x-23$

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Sub-topic 57 2020 · National One-Eighth Premium

Maximum of $f(x)=-x^2+6x-11$

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Sub-topic 58 2020 · National One-Eighth Premium

Minimum of $g(x)=3x^2-12x+16$

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Sub-topic 59 2020 · National One-Eighth Premium

Maximum of $h(x)=-2x^2-12x-15$

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Sub-topic 60 2021 · National Preliminary Premium

Quadratic equation with roots $\sqrt{5}\pm2$

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Sub-topic 61 2021 · National Preliminary Premium

Solution set of $2x^2-3x-5\gt0$

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Sub-topic 62 2021 · National Quarterfinal Premium

$\dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}$ for the roots of $x^2-2x+3=0$

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Sub-topic 63 2022 · National Preliminary Premium

Solution set of $x^2+2x-15\gt0$

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Sub-topic 64 2022 · National Preliminary Premium

Solution set of $x^2-3x-10\lt0$

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Sub-topic 65 2022 · National Preliminary Premium

Solution set of $x^2-7x+12\gt0$

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Sub-topic 66 2022 · National Preliminary Premium

Range of $y=-x^2+6x+7$

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Sub-topic 67 2022 · National Preliminary Premium

Real numbers $a$, $b$ with $a+b=5$ and $ab=5$

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Sub-topic 68 2022 · National Preliminary Premium

Real numbers $a$, $b$ with $a+b=-7$ and $ab=9$

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Sub-topic 69 2022 · National Preliminary Premium

Real numbers $a$, $b$ with $a+b=3$ and $ab=-3$

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Sub-topic 70 2022 · National Preliminary Premium

$(\alpha-\beta)^2$ for the complex roots of $x^2-5x+7=0$

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Sub-topic 71 2022 · National Preliminary Premium

Parabola through $(1,2)$, $(-1,4)$ and $(2,4)$ and its intercepts

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Sub-topic 72 2023 · National Preliminary Premium

Is $x^3+27=(x-3)(x^2+3x-9)$? (true/false)

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Sub-topic 73 2023 · National Preliminary Premium

Is $x^3-125=(x-5)(x^2+5x+25)$? (true/false)

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Sub-topic 74 2023 · National Preliminary Premium

Is $x^3+64=(x+4)(x^2-4x+16)$? (true/false)

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Sub-topic 75 2023 · National Preliminary Premium

Finding $b$ and $c$ when $x^2+bx+c\lt0$ has solution set $-1\lt x\lt5$

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Sub-topic 76 2023 · National Preliminary Premium

Is $x+2$ a factor of $x^3+4x^2+6x-2$? (true/false)

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Sub-topic 77 2023 · National Preliminary Premium

Is $x+3$ a factor of $x^3+3x^2+3x+9$? (true/false)

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Sub-topic 78 2023 · National Preliminary Premium

Is $x+4$ a factor of $x^3+4x^2+2x+8$? (true/false)

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Sub-topic 79 2023 · National Preliminary Premium

Least value of $x^2-6x+10$ and the corresponding $x$

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Sub-topic 80 2023 · National Preliminary Premium

Solution set of $2x^2-x-3\lt0$

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Quadratic equations and functions

Solving $(x-2)^{x-1}=1$, $(x+2)^{x-2}=1$ and $(2x-1)^{x-3}=1$

Least and greatest values of $x^2+6x-7$ on $2\leq x\leq5$ (vertex outside the interval)

Factorised equation of the parabola with $x$-intercepts $1$, $3$ and $y$-intercept $9$

Factorised equation of a parabola from its intercepts, e.g. $x=\pm2$ and $y=-8$

Riddle: the quadratic equation whose roots have sum $3\times5$ and product $3+5$

Quadratic equation with integer coefficients having the root $3+\sqrt{7}$

Quadratic equation with integer coefficients having the root $2-\sqrt{7}$

Factorised equation with $x$-intercepts $-3$, $5$ and $y$-intercept $30$ (speed race)

Is $\alpha^2+\beta^2=b^2+4c$ for the roots of $x^2+bx+c=0$? (true/false)

Least and greatest values of $x^2+6x+12$ on $-4\leq x\leq2$ (vertex inside the interval)

$(\alpha-\beta)^2$ for the roots of $x^2-3x+5=0$

Constants $a$, $b$, $c$ in $a-b(x+c)^2$ equal to a given quadratic

Constants in $a(x-3)^2+b(x-3)+c=2x^2-5x-12$

Factor theorem: is a quadratic a factor of a cubic polynomial? (true/false)

Solving $x+y=8$ and $y=x^2-12$

Constants in $3x^2+12x+20=a(x+b)^2+c$

Compound inequality $x+3\lt x^2+1\lt10$

Solving $|x^2-2|=x$

Constants $p$ and $q$ when $x^2-2x+3$ divides $3x^3+px^2+qx+15$

Maximum product of two numbers with sum $16$

Compound inequality $16\lt x^2\lt36$

Solving $|x^2-30|=x$

$x^2+xy+y^2$ from $x^3-y^3=56$ and $x-y=4$

Vertex of $y=3x^2-6x+7$

Compound inequalities such as $1\lt x^2-3\lt33$

Solving absolute value equations such as $|x^2-3x|=4$

Constants $p$ and $q$ when $px^2+qx-8\lt0$ has solution set $-2\lt x\lt4$

Solving $(x-1)(x+2)\gt4$

Factorised equation with intercepts $(-2,0)$, $(3,0)$ and $(0,-12)$

Values of $k$ for which $3x^2+2kx+2k=0$ has no real roots

Values of $k$ for which $x^2+2kx+16$ is positive for all real $x$

Greatest value of $-6+12x-x^2$ and the corresponding $x$

Maximum value of $25+10x-x^2$ and the corresponding $x$

Solving $(x+4)(x-3)\lt8$

Constants in $10-12x-2x^2=a-b(x+c)^2$

Solving $(x-5)(x+2)\lt18$

Curve with a stationary point on the $x$-axis at $a(1,0)$ and $y$-intercept $b(0,6)$

Nature of the roots of $x^2+(k-3)x+k=0$ for varying $k$

$(\alpha+\beta)^2$ for the roots of $x^2-3x+5=0$

Sum, product and nature of the roots of a quadratic (true/false)

Inverse variation: $y\propto\dfrac{1}{x^2}$ with $y=1$ when $x=5$

Direct variation: $y\propto x^3$ with $y=1$ when $x=-2$

Minimum sum of two positive numbers with product $100$

Maximum product of two numbers with sum $10$

Symmetric functions of the roots of $x^2+bx+c=0$ and a derived quadratic

Riddle: a three-digit number built from the roots of $x^2-3x+5=0$

Completing the square: $4x^2-16x+21=a(x+b)^2+c$

Completing the square: $3x^2+18x+10=a(x+b)^2+c$

Completing the square: $2x^2-20x+37=a(x+b)^2+c$

Does $x^2+5x-4=0$ have real roots? (true/false)

Does $x^2+5x+4=0$ have rational roots? (true/false)

Does $x^2+x+2=0$ have no real roots? (true/false)

Maximum or minimum point of $y=3+2x-x^2$

Maximum or minimum point of $y=2x^2-4x+5$

Maximum or minimum point of $y=x^2+4x+5$

Vertex of $y=-3x^2+18x-23$

Maximum of $f(x)=-x^2+6x-11$

Minimum of $g(x)=3x^2-12x+16$

Maximum of $h(x)=-2x^2-12x-15$

Quadratic equation with roots $\sqrt{5}\pm2$

Solution set of $2x^2-3x-5\gt0$

$\dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}$ for the roots of $x^2-2x+3=0$

Solution set of $x^2+2x-15\gt0$

Solution set of $x^2-3x-10\lt0$

Solution set of $x^2-7x+12\gt0$

Range of $y=-x^2+6x+7$

Real numbers $a$, $b$ with $a+b=5$ and $ab=5$

Real numbers $a$, $b$ with $a+b=-7$ and $ab=9$

Real numbers $a$, $b$ with $a+b=3$ and $ab=-3$

$(\alpha-\beta)^2$ for the complex roots of $x^2-5x+7=0$

Parabola through $(1,2)$, $(-1,4)$ and $(2,4)$ and its intercepts

Is $x^3+27=(x-3)(x^2+3x-9)$? (true/false)

Is $x^3-125=(x-5)(x^2+5x+25)$? (true/false)

Is $x^3+64=(x+4)(x^2-4x+16)$? (true/false)

Finding $b$ and $c$ when $x^2+bx+c\lt0$ has solution set $-1\lt x\lt5$

Is $x+2$ a factor of $x^3+4x^2+6x-2$? (true/false)

Is $x+3$ a factor of $x^3+3x^2+3x+9$? (true/false)

Is $x+4$ a factor of $x^3+4x^2+2x+8$? (true/false)

Least value of $x^2-6x+10$ and the corresponding $x$