Quadratic in exponential form

Solve equations that reduce to quadratics by substituting u = a^x or u = e^x, such as (a^x)² + pa^x + q = 0.

26 questions · Moderate -0.3

1.06g Equations with exponentials: solve a^x = b
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CAIE P2 2020 Specimen Q4
8 marks Moderate -0.3
4
  1. Solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving your answer correct to 3 significant figures.
  2. It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.
CAIE P2 2010 June Q5
6 marks Moderate -0.8
5
  1. Given that \(y = 2 ^ { x }\), show that the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ can be written in the form $$y ^ { 2 } - 4 y + 3 = 0$$
  2. Hence solve the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ giving the values of \(x\) correct to 3 significant figures where appropriate.
CAIE P2 2012 June Q2
5 marks Moderate -0.8
2
  1. Given that \(5 ^ { 2 x } + 5 ^ { x } = 12\), find the value of \(5 ^ { x }\).
  2. Hence, using logarithms, solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving the value of \(x\) correct to 3 significant figures.
CAIE P2 2016 June Q3
6 marks Moderate -0.3
3 Given that \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\), find the possible values of \(\mathrm { e } ^ { x }\) and hence solve the equation \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\) correct to 3 significant figures.
CAIE P2 2018 June Q1
5 marks Moderate -0.3
1 Solve the equation \(3 \mathrm { e } ^ { 2 x } - 82 \mathrm { e } ^ { x } + 27 = 0\), giving your answers in the form \(k \ln 3\).
CAIE P3 2015 June Q2
4 marks Moderate -0.8
2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.
CAIE P3 2017 June Q3
4 marks Moderate -0.8
3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.
CAIE P3 2019 June Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(9 ^ { x } = 3 ^ { x } + 12\). Give your answer correct to 2 decimal places.
CAIE P2 2002 November Q3
6 marks Moderate -0.8
3
  1. Express \(9 ^ { x }\) in terms of \(y\), where \(y = 3 ^ { x }\).
  2. Hence solve the equation $$2 \left( 9 ^ { x } \right) - 7 \left( 3 ^ { x } \right) + 3 = 0 ,$$ expressing your answers for \(x\) in terms of logarithms where appropriate.
CAIE P2 2011 November Q4
5 marks Moderate -0.3
4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.
CAIE P2 2018 November Q2
5 marks Moderate -0.3
2 Given that \(9 ^ { x } + 3 ^ { x } = 240\), find the value of \(3 ^ { x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.
CAIE P3 2020 June Q3
6 marks Standard +0.3
3
  1. Show that the equation $$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$ can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
  2. Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.
CAIE P3 2021 June Q2
5 marks Standard +0.3
2 Solve the equation \(4 ^ { x } = 3 + 4 ^ { - x }\). Give your answer correct to 3 decimal places.
CAIE P3 2023 June Q1
3 marks Standard +0.3
1 Solve the equation $$3 \mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { - 2 x } = 5$$ Give the answer correct to 3 decimal places.
Edexcel C2 2008 June Q4
6 marks Moderate -0.3
4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).
OCR C2 Q5
9 marks Moderate -0.3
5. (i) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (ii) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$
Edexcel PMT Mocks Q8
4 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-14_976_1296_283_429} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curves with equation \(y = 21 - 2 ^ { x }\) meet the curve with equation \(y = 2 ^ { 2 x + 1 }\) at the point \(A\) as shown in Figure 2. Find the exact coordinates of point \(A\).
Edexcel Paper 2 2020 October Q5
4 marks Standard +0.3
  1. The curve with equation \(y = 3 \times 2 ^ { x }\) meets the curve with equation \(y = 15 - 2 ^ { x + 1 }\) at the point \(P\). Find, using algebra, the exact \(x\) coordinate of \(P\).
Edexcel C2 Q6
9 marks Moderate -0.3
  1. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for
    1. \(3 ^ { x + 1 }\),
    2. \(3 ^ { 2 x - 1 }\).
      (b) Hence, or otherwise, solve the equation
    $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$ giving non-exact answers to 2 decimal places.
AQA C3 2005 June Q5
7 marks Moderate -0.3
5
  1. Solve the equation \(2 \mathrm { e } ^ { x } = 5\), giving your answer as an exact natural logarithm.
    1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\) can be written as $$2 y ^ { 2 } - 7 y + 5 = 0$$
    2. Hence solve the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\), giving your answers as exact values of \(x\).
AQA C3 2009 January Q7
6 marks Moderate -0.3
7
  1. Given that \(3 \mathrm { e } ^ { x } = 4\), find the exact value of \(x\).
    1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\) can be written as \(3 y ^ { 2 } - 19 y + 20 = 0\).
    2. Hence solve the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\), giving your answers as exact values. (3 marks)
CAIE P3 2024 November Q4
3 marks Moderate -0.3
Solve the equation \(5^x = 5^{x+2} - 10\). Give your answer correct to 3 decimal places. [3]
CAIE P3 2013 November Q2
4 marks Standard +0.3
Solve the equation \(2|3^x - 1| = 3^x\), giving your answers correct to 3 significant figures. [4]
OCR C1 Q6
10 marks Moderate -0.8
\begin{enumerate}[label=(\alph*)] \item Given that \(y = 2^x\), find expressions in terms of \(y\) for
  1. \(2^{x+2}\), [2]
  2. \(2^{3-x}\). [2]
\item Show that using the substitution \(y = 2^x\), the equation $$2^{x+2} + 2^{3-x} = 33$$ can be rewritten as $$4y^2 - 33y + 8 = 0.$$ [2] \item Hence solve the equation $$2^{x+2} + 2^{3-x} = 33.$$ [4]
Edexcel C2 Q3
6 marks Moderate -0.3
  1. Using the substitution \(u = 2^x\), show that the equation \(4^x - 2^{(x + 1)} - 15 = 0\) can be written in the form \(u^2 - 2u - 15 = 0\). [2]
  2. Hence solve the equation \(4^x - 2^{(x + 1)} - 15 = 0\), giving your answers to 2 d. p. [4]