Completing the square and sketching

Expand and simplify \((n + 2)^3 - n^3\). [3]

1 Simplify \(( 2 x + 5 ) ^ { 2 } - ( x - 3 ) ^ { 2 }\), giving your answer in the form \(a x ^ { 2 } + b x + c\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$

1. Write \(\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a , b\) and \(c\) are constants to be found. The point \(M\) is the minimum turning point of \(C\), as shown in Figure 3.
2. Deduce the coordinates of \(M\) The line \(l\) is the normal to \(C\) at the point \(P\), as shown in Figure 3.
   Given that \(l\) has equation \(y = k - \frac { 1 } { 8 } x\), where \(k\) is a constant,
3. 1. find the coordinates of \(P\)
   2. find the value of \(k\) Question 9 continues on the next page \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 is a copy of Figure 3. The finite region \(R\), shown shaded in Figure 4, is bounded by \(l , C\) and the line through \(M\) parallel to the \(y\)-axis.
4. Identify the inequalities that define \(R\).

Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]

1. Given that

$$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$

1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
   • meets the \(y\)-axis at the point \(P\)
   • has a minimum turning point at the point \(Q\)
   • Write down
     1. the coordinates of \(P\)
     2. the coordinates of \(Q\)

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = - 3 x ^ { 2 } + 12 x + 8$$

1. Write \(\mathrm { f } ( x )\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The curve \(C\) has a maximum turning point at \(M\).
2. Find the coordinates of \(M\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-34_735_841_913_612} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve \(C\).
   The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
3. Using algebraic integration, find the area of \(R\).

Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]

7

1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.

9

1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).

1. Solve the equation

$$x ^ { 2 } - 4 x - 8 = 0$$ giving your answers in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.

1. Solve the equation \(x^2 - 8x + 4 = 0\), giving your answer in the form \(p \pm q\sqrt{3}\), where \(p\) and \(q\) are integers. [2]
2. Expand and simplify \((6 + 2\sqrt{3})(2 - \sqrt{3})\). [3]

4. (i) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$ (ii) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$

9

1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.

9

1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.

$$f(x) = x^2 - 10x + 17.$$

1. Express \(f(x)\) in the form \(a(x + b)^2 + c\). [3]
2. State the coordinates of the minimum point of the curve \(y = f(x)\). [1]
3. Deduce the coordinates of the minimum point of each of the following curves:
   1. \(y = f(x) + 4\), [2]
   2. \(y = f(2x)\). [2]

Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]

Rearrange the following equation to make \(h\) the subject. $$4h + 5 = 9a - ha^2$$ [3]

\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2\(x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

1. Express \(2x^2 - 5x + k\) in the form \(a(x - b)^2 + c\) [3 marks]
2. Find the values of \(k\) for which the curve \(y = 2x^2 - 5x + k\) does not intersect the line \(y = 3\) [3 marks]

1. 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 \(p\). [2]
2. Hence or otherwise find the set of values of \(p\) for which the equation \(4x^2 - 24x + p = 0\) has no real roots. [1]

$$f(x) = x^3 - (k+4)x + 2k,$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve with equation \(y = f(x)\) passes through the point \((2, 0)\). [1]
2. Find the values of \(k\) for which the equation \(f(x) = 0\) has exactly two distinct roots. [5]

Given that \(k > 0\), that the \(x\)-axis is a tangent to the curve with equation \(y = f(x)\), and that the line \(y = p\) intersects the curve in three distinct points,

1. find the set of values that \(p\) can take. [5]

1. The equation \(x^2 + 3x + k = 0\) has repeated roots. Find the value of the constant \(k\). [2]
2. Solve the inequality \(6 + x - x^2 > 0\). [2]

1. 1. Express \(x^2 - 8x + 11\) in the form \((x - a)^2 + b\) where \(a\) and \(b\) are constants. [2]
   2. Hence write down the minimum value of \(x^2 - 8x + 11\). [1]
2. Determine the value of the constant \(k\) for which the equation \(x^2 - 8x + 11 = k\) has two equal roots. [2]

7

1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).

Find the real roots of the equation \(4x^4 + 3x^2 - 1 = 0\). [5]

1. Show that the graph of \(y = x^2 - 3x + 11\) is above the \(x\)-axis for all values of \(x\). [3]
2. Find the set of values of \(x\) for which the graph of \(y = 2x^2 + x - 10\) is above the \(x\)-axis. [4]
3. Find algebraically the coordinates of the points of intersection of the graphs of $$y = x^2 - 3x + 11 \quad\text{and}\quad y = 2x^2 + x - 10.$$ [5]

9

1. Show that \(( x - 1 ) ( x - 2 ) ( x - 3 ) - \left( x ^ { 3 } - x ^ { 2 } + 11 x - 12 \right) = 6 - 5 x ^ { 2 }\).
2. Solve the equation \(6 - 5 x ^ { 2 } = 0\).

1. Express \(3y^2 - 12y - 15\) in the form \(3(y + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
2. Hence find the exact solutions of the equation \(3x^4 - 12x^2 - 15 = 0\). [3]

A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r\) cm.

1. Show that the area of the sector, \(A\) cm\(^2\), is given by \(A = 12r - r^2\). [3]
2. Express \(A\) in the form \(a - (r - b)^2\), where \(a\) and \(b\) are constants. [2]
3. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3x^2 + 6x + 10\) and the line \(y = 2 - 4x\). [5]
2. Write \(3x^2 + 6x + 10\) in the form \(a(x + b)^2 + c\). [4]
3. Hence or otherwise, show that the graph of \(y = 3x^2 + 6x + 10\) is always above the \(x\)-axis. [2]

1. Express \(x^2 - 6x + 2\) in the form \((x - a)^2 - b\). [3]
2. State the coordinates of the turning point on the graph of \(y = x^2 - 6x + 2\). [2]
3. Sketch the graph of \(y = x^2 - 6x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis. [2]
4. Solve the simultaneous equations \(y = x^2 - 6x + 2\) and \(y = 2x - 14\). Hence show that the line \(y = 2x - 14\) is a tangent to the curve \(y = x^2 - 6x + 2\). [5]

3 Given that \(3 x ^ { 2 } + b x + 10 = a ( x + 3 ) ^ { 2 } + c\) for all values of \(x\), find the values of the constants \(a , b\) and \(c\).

1 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h] \end{figure}

1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \({ } ^ { 2 }\).
3. Find the area of the region bounded by the curve and the \(x\)-axis.

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

2

1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
3. Find the value of the constant \(k\) for which the line \(y = k - 2 x\) is a tangent to the curve \(y = 2 x ^ { 2 } + 6 x + 5\).

Simplify \((m^2 + 1)^2 - (m^2 - 1)^2\), showing your method. Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. [4] \includegraphics{figure_3}

9 The graph shows the function \(y = x ^ { 2 } + b x + c\) where \(b\) and \(c\) are constants.
The point \(\mathrm { M } ( - 3 , - 16 )\) on the graph is the minimum point of the graph. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-2_478_948_1871_588}

1. Write down the function \(y = \mathrm { f } ( x )\) in completed square form.
2. Hence find the coordinates of the points where the curve cuts the axes.