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

Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]

Find the real roots of the equation \(x^4 - 5x^2 - 36 = 0\) by considering it as a quadratic equation in \(x^2\). [4]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).

1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]

The sequence \(u_1, u_2, u_3, ...\) is defined by the recurrence relation $$u_{n+1} = (u_n)^2 - 1, \quad n \geq 1.$$ Given that \(u_1 = k\), where \(k\) is a constant,

1. find expressions for \(u_2\) and \(u_3\) in terms of \(k\). [3]

Given also that \(u_2 + u_3 = 11\),

1. find the possible values of \(k\). [4]

Solve the simultaneous equations \begin{align} x + y &= 2
3x^2 - 2x + y^2 &= 2 \end{align} [7]

$$f(x) = 2x^2 + 3x - 2.$$

1. Solve the equation \(f(x) = 0\). [2]
2. Sketch the curve with equation \(y = f(x)\), showing the coordinates of any points of intersection with the coordinate axes. [2]
3. Find the coordinates of the points where the curve with equation \(y = f(\frac{1}{2}x)\) crosses the coordinate axes. [3]

When the graph of \(y = f(x)\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants.

1. Find the values of \(a\), \(b\) and \(c\). [3]

Solve the equation $$3x - \frac{5}{x} = 2.$$ [4]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 8x - x^{\frac{3}{2}}\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the \(x\)-coordinate of \(A\). [3]
2. Find the gradient of the tangent to the curve at \(A\). [4]

$$\text{f}(x) = 2x^2 - 4x + 1.$$

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\text{f}(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = \text{f}(x)\). [1]
3. Solve the equation \(\text{f}(x) = 3\), giving your answers in exact form. [3]

Find in exact form the real solutions of the equation $$x^4 = 5x^2 + 14.$$ [3]

\(\text{f}(x) = 4x^2 + 12x + 9\).

1. Determine the number of real roots that exist for the equation \(\text{f}(x) = 0\). [2]
2. Solve the equation \(\text{f}(x) = 8\), giving your answers in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are rational. [4]

\(f(x) = 2x^2 - 4x + 1\).

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$f(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = f(x)\). [1]
3. Solve the equation \(f(x) = 3\), giving your answers in exact form. [3]