1.02e Complete the square: quadratic polynomials and turning points

4. (a) Show that \(x ^ { 2 } + 6 x + 11\) can be written as $$( x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers to be found.
(b) In the space at the top of page 7 , sketch the curve with equation \(y = x ^ { 2 } + 6 x + 11\), showing clearly any intersections with the coordinate axes.
(c) Find the value of the discriminant of \(x ^ { 2 } + 6 x + 11\)

7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where \(k\) is a real constant.

1. Find the discriminant of \(\mathrm { f } ( x )\) in terms of \(k\).
2. Show that the discriminant of \(\mathrm { f } ( x )\) can be expressed in the form \(( k + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
3. Show that, for all values of \(k\), the equation \(\mathrm { f } ( x ) = 0\) has real roots.

8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where \(p\) and \(q\) are integers.

1. Find the value of \(p\) and the value of \(q\).
2. Calculate the discriminant of \(4 x - 5 - x ^ { 2 }\)
3. On the axes on page 17, sketch the curve with equation \(y = 4 x - 5 - x ^ { 2 }\) showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-11_1143_1143_260_388}

1. The equation

$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$ has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } - 6 p + 1 > 0\)
2. Hence find the set of possible values of \(p\).

5. $$f ( x ) = x ^ { 2 } - 8 x + 19$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) crosses the \(y\)-axis at the point \(P\) and has a minimum point at the point \(Q\).
2. Sketch the graph of \(C\) showing the coordinates of point \(P\) and the coordinates of point \(Q\).
3. Find the distance \(P Q\), writing your answer as a simplified surd.

3. $$f ( x ) = x ^ { 2 } - 10 x + 23$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants to be found.
2. Hence, or otherwise, find the exact solutions to the equation $$x ^ { 2 } - 10 x + 23 = 0$$
3. Use your answer to part (b) to find the larger solution to the equation $$y - 10 y ^ { 0.5 } + 23 = 0$$ Write your solution in the form \(p + q \sqrt { r }\), where \(p , q\) and \(r\) are integers.

2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2\).
2. Show that \(x ^ { 2 } + x + 1 > 0\) for all values of \(x\).
3. Show that \(\mathrm { f } ( x ) > 0\) for all values of \(x , x \neq - 2\).

$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b ,$$ where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\).
2. Hence, or otherwise, show that the roots of $$x ^ { 2 } - 8 x - 29 = 0$$ are \(c \pm d \sqrt { } 5\), where \(c\) and \(d\) are integers to be found.

5. $$4 x ^ { 2 } + 4 x + 17 \equiv ( 2 x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers.

1. Determine the value of \(p\) and the value of \(q\) Given that $$\frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \equiv \frac { 1 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } + \frac { A x + B } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } }$$ where \(A\) and \(B\) are integers,
2. write down the value of \(A\) and the value of \(B\)
3. Hence use algebraic integration to show that $$\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \mathrm {~d} x = k + \frac { 1 } { 2 } \ln k$$ where \(k\) is a rational number to be determined.

2. $$9 x ^ { 2 } + 6 x + 5 \equiv a ( x + b ) ^ { 2 } + c$$

1. Find the values of the constants \(a\), \(b\) and \(c\). Hence, or otherwise, find
2. \(\int \frac { 1 } { 9 x ^ { 2 } + 6 x + 5 } d x\)
3. \(\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 6 x + 5 } } \mathrm {~d} x\)

7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)

1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
5. Find the minimum area of triangle \(O A B\).

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   Q7

2 Given that \(2 x ^ { 2 } - 12 x + p = q ( x - r ) ^ { 2 } + 10\) for all values of \(x\), find the constants \(p , q\) and \(r\).

5

1. Express \(x ^ { 2 } + 3 x\) in the form \(( x + a ) ^ { 2 } + b\).
2. Express \(y ^ { 2 } - 4 y - \frac { 11 } { 4 }\) in the form \(( y + p ) ^ { 2 } + q\). A circle has equation \(x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0\).
3. Write down the coordinates of the centre of the circle.
4. Find the radius of the circle.

6

1. Express \(2 x ^ { 2 } - 24 x + 80\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } - 24 x + 80\).
3. State the equation of the tangent to the curve \(y = 2 x ^ { 2 } - 24 x + 80\) at its minimum point.

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 \(c\).

2

1. Express \(3 x ^ { 2 } + 12 x + 7\) in the form \(3 ( x + a ) ^ { 2 } + b\).
2. Hence write down the equation of the line of symmetry of the curve \(y = 3 x ^ { 2 } + 12 x + 7\).

8

1. Express \(x ^ { 2 } + 8 x + 15\) in the form \(( x + a ) ^ { 2 } - b\).
2. Hence state the coordinates of the vertex of the curve \(y = x ^ { 2 } + 8 x + 15\).
3. Solve the inequality \(x ^ { 2 } + 8 x + 15 > 0\).

10

1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).

2

1. Express \(x ^ { 2 } - 8 x + 3\) in the form \(( x + a ) ^ { 2 } + b\).
2. Hence write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 8 x + 3\).

11

1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

12

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

12

1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).

10

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

11 \begin{figure}[h] \end{figure} Fig. 11 shows a sketch of the circle with equation \(( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 125\) and centre C . The points \(\mathrm { A } , \mathrm { B }\), D and E are the intersections of the circle with the axes.

1. Write down the radius of the circle and the coordinates of C .
2. Verify that B is the point \(( 21,0 )\) and find the coordinates of \(\mathrm { A } , \mathrm { D }\) and E .
3. Find the equation of the perpendicular bisector of BE and verify that this line passes through C .

12

1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).