Mixed practice: equations Problem set B

Form a quadratic equation for each situation and solve it.

1. (a) The base of a triangle is 6 cm longer than its perpendicular height \(h\) cm. The area of the triangle is 17 cm². Find the height to 3 significant figures.
2. (b) Two numbers differ by 4. Their product is 9. The smaller number is \(x\). Write this as a quadratic equation \(ax^2 + bx + c = 0\) and find the possible values of \(x\) to 3 significant figures.

Rearrange and solve a quadratic.

1. (a) Show that \((2x − 3)(x + 4) = 16\) can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are integers. Write down \(a\), \(b\) and \(c\).
2. (b) Hence solve \((2x − 3)(x + 4) = 16\). Give the solutions to 3 significant figures.

Turn a fraction equation into a cubic and solve it.

1. (a) Show that \(x^2 − 6 = \dfrac{4}{x + 3}\) can be written in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\) and \(d\) are integers. Write down \(a\), \(b\), \(c\) and \(d\).
2. (b) Hence solve \(x^2 − 6 = \dfrac{4}{x + 3}\). Give the real solutions to 3 significant figures.

Form a polynomial equation and solve it with the calculator.

1. (a) The product of three consecutive integers is 2184. Find the mean of the three integers.
2. (b) Two numbers differ by 1. Their cubes differ by 8. Find the possible values of the smaller number to 3 significant figures.

Form a quadratic in \(n\) and solve it.

1. (a) The first term of an arithmetic sequence is 4 and the common difference is 5. The sum of the first \(n\) terms is 837. Find the value of \(n\).
2. (b) In a chess tournament, every player plays every other player once. There are 136 games in total. Find the number of players, \(n\).

A storage crate is a cuboid with edges of length \(x\) cm, \((x + 4)\) cm and \((3x + 1)\) cm, as shown.

1. (a) Write down an expression for the volume \(V\) of the cuboid. Expand and simplify it.
2. (b) The volume of the cuboid is 160 cm³. Find the value of \(x\) to 3 significant figures.
3. (c) The numerical values of the volume and the surface area of a cube are added together. The total is 200. Find the edge length of the cube to 3 significant figures.

A car park holds cars (4 wheels) and motorbikes (2 wheels). There are 25 vehicles with 78 wheels in total. Let \(c\) be the number of cars and \(m\) the number of motorbikes.

1. (a) Write down two equations in \(c\) and \(m\).
2. (b) Solve them with the calculator. How many cars are there?

A market stall sells small boxes and large boxes of strawberries. Let \(x\) be the number of large boxes and \(y\) the number of small boxes sold. A large box costs €16 and a small box costs €8. 2 500 boxes are sold.

1. (a) Write down an equation in \(x\) and \(y\) for the number of boxes sold.
2. (b) Find the cost of 2 large boxes and 5 small boxes.
3. (c) The total money from sales is €31 600. Write down a second equation in \(x\) and \(y\).
4. (d) Find the number of large boxes and the number of small boxes sold.

A forester plants trees in rows to make a triangle. The first row has 1 tree and each row has one more tree than the row before. A plantation has \(n\) trees in the last row and \(S\) in total.

1. (a) Show that a plantation with 30 trees in the last row has 465 trees in total.
2. (b) A plantation has 1 275 trees in total. Find the number of trees in the last row.
3. (c) Show that \(n^2 + n − 2S = 0\).
4. (d) A forester has 600 trees. Explain why they cannot be planted in this way with none left over.

A rectangular sheet of card measures \((x + 7)\) cm by \((x + 1)\) cm. A square of side 1.5 cm is cut from one corner. The card that is left is the shaded region.

1. (a) Write down an expression for the area \(A\) of the shaded region in terms of \(x\). Expand and simplify it.
2. (b) The area of the shaded region is 60 cm². Find \(x\) to 3 significant figures.
3. (c) Find the perimeter of the shaded region to 3 significant figures.

Use the sequence formulae and the calculator.

1. (a) In an arithmetic sequence, \(S_{24} = 1 284\) and \(u_{24} = 111\). Find \(u_1\) and \(d\).
2. (b) A geometric sequence has first term 5 and the sum of its first three terms is 7.2. Find the possible values of the common ratio \(r\), to 3 significant figures.

In triangle ABC, \(AB = x\), \(AC = (x + 2)\), \(BC = 3x\) and angle \(BAC = 120°\).

1. (a) Show that \(3 x^{2} - 3 x - 2 = 0\).
2. (b) Hence find the value of \(x\) to 3 significant figures.

Form equations and solve them with the calculator.

1. (a) The curve \(y = ax^2 + bx\) passes through the point \((2, -2)\) and has gradient 5 at that point. Find the value of \(y\) when \(x = 4\).
2. (b) Solve the simultaneous equations \(x^3 + y^3 + z^3 = 21,\ 3x^3 + y^3 − z^3 = 13,\ x^3 − 2y^3 + 3z^3 = −62\).

Every member of a club of 60 does swimming or cycling or both. The number who swim is 4 more than the number who cycle. The number who do both is 16 less than the number who do only one of the two activities.

1. (a) Let \(h\) be the number who do swimming only, \(t\) the number who do cycling only and \(b\) the number who do both. Write down three equations in \(h\), \(t\) and \(b\).
2. (b) Solve the equations with the calculator. A person is chosen at random. Find the probability that they do both.