Lesson Guide
Explanation
Let the width be w centimetres. The length is three more, so it is w plus three. The area is width times length, so w times w plus three equals forty. Expand and subtract forty: w squared plus three w minus forty equals zero. Factorise: w plus eight, times w minus five, equals zero. So w is negative eight or five. But a width cannot be negative, so you reject negative eight and say why. The rectangle is five centimetres by eight centimetres, and five times eight is forty.
Your Turn
A rectangle’s length is 6 cm more than its width and its area is 55 cm2. Check: w2 + 6w − 55 = 0, (w + 11)(w − 5) = 0, so 5 cm by 11 cm.
Watch Out
w = −8 solves the equation but not the problem. Reject it in words: "a width cannot be negative".
Explanation
Two consecutive integers multiply to seventy-two. Let the smaller one be n, so the next one is n plus one. Then n times n plus one equals seventy-two, which becomes n squared plus n minus seventy-two equals zero. Factorise: n plus nine, times n minus eight. So n is eight or negative nine. This time, do not throw the negative one away. Integers can be negative, and negative nine times negative eight is also seventy-two. So there are two pairs: eight and nine, or negative nine and negative eight.
Your Turn
Two consecutive even integers multiply to 80. Check: n(n + 2) = 80, (n + 10)(n − 8) = 0, so 8 and 10, or −10 and −8.
Watch Out
Do not reject a negative root out of habit. If the problem says "integers", negative ones count; only "positive" or a length rules them out.
Explanation
A garden six metres by four metres has a path of the same width, x metres, all the way round it. The garden and path together cover forty-eight square metres. The path is on both sides, so the outer rectangle is six plus two x by four plus two x. Multiply out: twenty-four plus twenty x plus four x squared equals forty-eight. Subtract forty-eight and divide everything by four: x squared plus five x minus six equals zero. That factorises to x plus six, times x minus one. A path cannot be negative six metres wide, so the path is one metre wide. Check: eight times six is forty-eight.
Your Turn
A 10 m by 6 m lawn has a path x m wide all round; together they cover 96 m2. Check: x2 + 8x − 9 = 0, so x = 1 m (12 × 8 = 96).
Watch Out
Writing (6 + x)(4 + x) puts the path on one side only. A path all round adds 2x to each length.
Explanation
A triangle's base is four centimetres more than its height, and its area is seventy square centimetres. Let the height be h, so the base is h plus four. The area is a half times base times height, so a half of h times h plus four equals seventy. Multiply both sides by two first: h times h plus four equals one hundred and forty. That gives h squared plus four h minus one hundred and forty equals zero, which factorises to h plus fourteen, times h minus ten. A height cannot be negative fourteen, so the height is ten and the base is fourteen. Check: half of ten times fourteen is seventy.
Your Turn
A triangle’s height is 3 cm less than its base and its area is 35 cm2. Check: b(b − 3) = 70, (b − 10)(b + 7) = 0, so base 10 cm and height 7 cm.
Watch Out
Do not forget the ½. h(h + 4) = 70 has no whole-number answer: if it will not factorise, look for a slip.
Explanation
Here is the whole method. Name the unknown. Write the equation. Bring everything to one side so it equals zero. Factorise and solve. Then test each root in the words of the problem, and keep or reject it with a reason. A number is twenty less than its square. Let it be n: n squared minus twenty equals n, so n squared minus n minus twenty equals zero. That is n minus five, times n plus four. Test both: five squared minus twenty is five, and negative four squared minus twenty is negative four. Nothing rules either out, so both are answers. Quiz and exercises now.
Your Turn
A positive number times 5 more than itself is 84. Check: n2 + 5n − 84 = 0, (n + 12)(n − 7) = 0, so n = 7 (−12 is not positive).
Watch Out
Answer the question asked. If it wants the length, giving only the width is not finished.
📌 Key Points — this lesson in a nutshell
- Name the unknown and write every other length in terms of it: "Let the width be w cm, so the length is w + 3 cm."
- Form the equation, expand, bring everything to one side (= 0), take out any common factor, then factorise.
- Test each root against the story. Reject a root that gives a negative length or breaks a condition — and write why.
- Do not reject automatically: integers can be negative, so both roots may be valid answers.
- Answer the question that was asked, with units, and check it in the words of the problem.
Step 1: Introduction
Step 1 of 5
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