Quadratics: past paper questions

Where the questions come from

• Question 1Oct/Nov 2025, paper 11Structured4 marks

  Find the set of values of the constant $k$ for which the quadratic equation $$3k{x}^2+(k+8)x+3=0$$ has two distinct real roots.

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• Question 4Oct/Nov 2024, paper 11Structured5 marks

  Show that the curve with equation $x^2-3xy-40=0$ and the line with equation $3x+y+k=0$ meet for all values of the constant $k$.

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• Question 2Oct/Nov 2023, paper 11Structured4 marks

  A line has equation $y=2cx+3$ and a curve has equation $y=c{x}^2+3x-c$, where $c$ is a constant. Showing all necessary working, determine which of the following statements is correct.

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• Question 4Oct/Nov 2025, paper 11Structured6 marks

  Express $1-6x-x^2$ in the form $a-(x+b{)}^2$, where $a$ and $b$ are constants.

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• Question 8Oct/Nov 2023, paper 11Structured9 marksHas a diagram

  The diagram shows the curves with equations $y=2(2x-3{)}^4$ and $y=(2x-3{)}^2+1$ meeting at points $A$ and $B$. By using the substitution $u=2x-3$ find, by calculation, the coordinates of $A$ and $B$.

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• Question 1May/June 2026, paper 12Structured3 marksHas a diagram

  Solve the equation $$3x^2-8-\frac{16}{x^2}=0.$$ Solve the equation $$3x^2-8-\frac{16}{x^2}=0.$$

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• Question 1Oct/Nov 2025, paper 12Structured5 marks

  Express $9x^2-36x+8$ in the form $p(x+q{)}^2+r$, where $p$, $q$ and $r$ are constants.

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• Question 7Oct/Nov 2024, paper 12Structured8 marksHas a diagram

  By expressing $-2x^2+8x+11$ in the form $-a(x-b{)}^2+c$, where $a$, $b$ and $c$ are positive integers, find the coordinates of the vertex of the graph with equation $y=-2x^2+8x+11$.

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• Question 9Oct/Nov 2023, paper 11Structured9 marks

  Express $4x^2-12x+13$ in the form $(2x+a{)}^2+b$, where $a$ and $b$ are constants.

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• Question 6Oct/Nov 2023, paper 12Structured8 marks

  The equation of a curve is $y=x^2-8x+5$. Find the coordinates of the minimum point of the curve.

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• Question 6May/June 2025, paper 11Structured9 marks

  The equation of a curve is $2x^2-kxy+2=0$ and the equation of a line is $y=px+3$, where $k$ and $p$ are constants. Given that $k=2$ and $p=11$, find the coordinates of the points of intersection of the curve and the line.

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• Question 6Oct/Nov 2023, paper 13Structured7 marks

  A line has equation $y=6x-c$ and a curve has equation $y=c{x}^2+2x-3$, where $c$ is a constant. The line is a tangent to the curve at point $P$. Find the possible values of $c$ and the corresponding coordinates of $P$.

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• Question 1May/June 2026, paper 13Structured5 marks

  Find the coordinates of the points of intersection of the curve $y=3x^2-12x+17$ and the line $y=6x+2$.

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• Question 8Oct/Nov 2024, paper 13Structured9 marksHas a diagram

  Express $3x^2-12x+14$ in the form $3(x+a{)}^2+b$, where $a$ and $b$ are constants to be found.

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• Question 5May/June 2023, paper 11Structured5 marks

  The line with equation $y=kx-k$, where $k$ is a positive constant, is a tangent to the curve with equation $y=-\frac{1}{2x}$. Find, in either order, the value of $k$ and the coordinates of the point where the tangent meets the curve.

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• Question 2May/June 2026, paper 13Structured4 marks

  Express $x\sqrt{x}$ in the form $x^n$, where $n$ is a rational number.

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• Question 11May/June 2025, paper 12Structured9 marks

  Express $x^2+4x+2$ in the form $(x+a{)}^2+b$, where $a$ and $b$ are integers.

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• Question 1May/June 2024, paper 11Structured5 marksHas a diagram

  Express $3y^2-12y-15$ in the form $3(y+a{)}^2+b$, where $a$ and $b$ are constants.

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• Question 3May/June 2023, paper 12Structured3 marks

  Express $4x^2-24x+p$ in the form $a(x+b{)}^2+c$, where $a$ and $b$ are integers and $c$ is to be given in terms of the constant $p$.

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• Question 4May/June 2023, paper 12Structured3 marks

  Solve the equation $8x^6+215x^3-27=0$.

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