1.02f Solve quadratic equations: including in a function of unknown

448 questions

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CAIE P2 2016 November Q7
12 marks Moderate -0.3
The polynomial \(p(x)\) is defined by $$p(x) = ax^3 + 3x^2 + bx + 12,$$ where \(a\) and \(b\) are constants. It is given that \((x + 3)\) is a factor of \(p(x)\). It is also given that the remainder is 18 when \(p(x)\) is divided by \((x + 2)\).
  1. Find the values of \(a\) and \(b\). [5]
  2. When \(a\) and \(b\) have these values,
    1. show that the equation \(p(x) = 0\) has exactly one real root, [4]
    2. solve the equation \(p(\sec y) = 0\) for \(-180° < y < 180°\). [3]
CAIE P2 2016 November Q1
5 marks Moderate -0.3
The sequence of values given by the iterative formula $$x_{n+1} = \frac{4}{x_n^2} + \frac{2x_n}{3},$$ with initial value \(x_1 = 2\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
  2. State an equation that is satisfied by \(\alpha\), and hence find the exact value of \(\alpha\). [2]
CAIE P3 2021 March Q3
6 marks Standard +0.3
By first expressing the equation \(\tan(x + 45°) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0° < x < 180°\). [6]
Edexcel P1 2018 Specimen Q4
5 marks Standard +0.3
The straight line with equation \(y = 4x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = 2x^2 + 8x + 3\) Calculate the value of \(c\) [5]
Edexcel C1 Q3
4 marks Moderate -0.8
Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]
Edexcel C1 Q10
12 marks Moderate -0.3
Given that $$f(x) = x^2 - 6x + 18, \quad x \geq 0,$$
  1. express \(f(x)\) in the form \((x - a)^2 + b\), where \(a\) and \(b\) are integers. [3]
The curve \(C\) with equation \(y = f(x)\), \(x \geq 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
  1. Sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). [4]
The line \(y = 41\) meets \(C\) at the point \(R\).
  1. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q\sqrt{2}\), where \(p\) and \(q\) are integers. [5]
Edexcel C1 Q8
6 marks Moderate -0.8
The equation \(x^2 + 2px + (3p + 4) = 0\), where \(p\) is a positive constant, has equal roots.
  1. Find the value of \(p\). [4]
  2. For this value of \(p\), solve the equation \(x^2 + 2px + (3p + 4) = 0\). [2]
Edexcel C1 Q4
7 marks Moderate -0.5
Solve the simultaneous equations $$y = x - 2,$$ $$y^2 + x^2 = 10.$$ [7]
Edexcel C1 Q14
5 marks Moderate -0.8
Given that \(f(x) = 15 - 7x - 2x^2\),
  1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
  2. Sketch the graph of \(y = f(x)\). [2]
Edexcel C1 Q15
8 marks Moderate -0.3
  1. By completing the square, find in terms of \(k\) the roots of the equation $$x^2 + 2kx - 7 = 0.$$ [4]
  2. Prove that, for all values of \(k\), the roots of \(x^2 + 2kx - 7 = 0\) are real and different. [2]
  3. Given that \(k = \sqrt{2}\), find the exact roots of the equation. [2]
Edexcel C1 Q30
7 marks Moderate -0.8
  1. Solve the equation \(4x^2 + 12x = 0\). [3]
\(f(x) = 4x^2 + 12x + c\), where \(c\) is a constant.
  1. Given that \(f(x) = 0\) has equal roots, find the value of \(c\) and hence solve \(f(x) = 0\). [4]
Edexcel C1 Specimen Q6
9 marks Moderate -0.8
  1. Solve the simultaneous equations $$y + 2x = 5,$$ $$2x^2 - 3x - y = 16.$$ [6]
  2. Hence, or otherwise, find the set of values of \(x\) for which $$2x^2 - 3x - 16 > 5 - 2x$$ [3]
Edexcel C2 Q9
10 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = -2x^2 + 4x\) and the line \(y = \frac{3}{2}\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find
  1. the \(x\)-coordinates of the points \(A\) and \(B\), [4]
  2. the exact area of \(R\). [6]
Edexcel C2 Q10
11 marks Standard +0.3
  1. Given that $$(2 + x)^5 + (2 - x)^5 = A + Bx^2 + Cx^4,$$ find the values of the constants \(A\), \(B\) and \(C\). [6]
  2. Using the substitution \(y = x^2\) and your answers to part (a), solve, $$(2 + x)^5 + (2 - x)^5 = 349.$$ [5]
Edexcel C2 Q23
8 marks Moderate -0.8
Given that \(f(x) = 15 - 7x - 2x^2\),
  1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
  2. Sketch the graph of \(y = f(x)\). [2]
  3. Calculate the coordinates of the stationary point of \(f(x)\). [3]
Edexcel C2 Q34
6 marks Moderate -0.8
  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 decimals places. [4]
Edexcel FP2 Q23
10 marks Challenging +1.2
  1. Use de Moivre's theorem to show that $$\cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.$$ [6]
  2. Hence find \(3\) distinct solutions of the equation \(16x^5 - 20x^3 + 5x + 1 = 0\), giving your answers to \(3\) decimal places where appropriate. [4]
Edexcel M2 2014 January Q3
7 marks Moderate -0.5
Solve the simultaneous equations $$x - 2y - 1 = 0$$ $$x^2 + 4y^2 - 10x + 9 = 0$$ [7]
Edexcel M2 2013 June Q7
16 marks Standard +0.3
\includegraphics{figure_4} A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9a\) from \(O\) and at a height \(6a\) above the level of \(O\). The ball is projected with speed \(\sqrt{(27ag)}\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.
  1. Show that tan\(^2 \theta - 6\) tan \(\theta + 5 = 0\) [7]
The two possible angles of projection are \(\theta_1\) and \(\theta_2\), where \(\theta_1 > \theta_2\).
  1. Find tan \(\theta_1\) and tan \(\theta_2\). [3]
The particle is projected at the larger angle \(\theta_1\).
  1. Show that the time of flight from \(O\) to \(T\) is \(\sqrt{\left(\frac{78a}{g}\right)}\). [3]
  2. Find the speed of the particle immediately before it hits \(T\). [3]
OCR C1 2013 January Q1
5 marks Moderate -0.8
  1. Solve the equation \(x^2 - 6x - 2 = 0\), giving your answers in simplified surd form. [3]
  2. Find the gradient of the curve \(y = x^2 - 6x - 2\) at the point where \(x = -5\). [2]
OCR C1 2006 June Q3
7 marks Moderate -0.8
  1. Express \(2x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). [4]
  2. Solve \(2x^2 + 12x + 13 = 0\), giving your answers in simplified surd form. [3]
OCR C1 2006 June Q6
8 marks Moderate -0.3
  1. Solve the equation \(x^4 - 10x^2 + 25 = 0\). [4]
  2. Given that \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\), find \(\frac{dy}{dx}\). [2]
  3. Hence find the number of stationary points on the curve \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\). [2]
OCR C1 2013 June Q2
5 marks Standard +0.3
Solve the equation \(8x^6 + 7x^3 - 1 = 0\). [5]
OCR C1 2014 June Q3
5 marks Standard +0.3
Find the real roots of the equation \(4x^4 + 3x^2 - 1 = 0\). [5]
OCR MEI C1 2006 January Q9
5 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
  2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]