The Ten-Euro Note — Solving a Quadratic

STEP 1 · PART (a)

Name the sides with algebra

Why we do this: we can't solve anything until every side has a name. We're told the width but not its number, so we call the width x and build the length from it.

The problem says the length is 6 cm longer than the width. "Longer than" just means add. So we take the width and add 6.

The idea (in words)

length = width + 6

widthwe don't know it, so we call it x

+ 6the length is 6 cm bigger than the width

↓ now swap "width" for x ↓

Put it in symbols length = x + 6

Answer (a)length = x + 6

STEP 2 · PART (b)

Build the equation

Why an area formula: the only other fact we're given is the area (90 cm²). Area links the two sides together — so writing the area in terms of x gives us one equation with one unknown, which we can solve.

Start with the raw formula

Area = length × width

Areathe space inside the rectangle = 90 cm²

lengthwe found this in Step 1: x + 6

widththat's just x

↓ put our parts in ↓

Substitute, then tidy up 90 = (x + 6) × x 90 = x² + 6x ← multiply the bracket out → x² + 6x − 90 = 0 ← move 90 across

Why move the 90 across? To solve a quadratic, we need it to equal zero. That "= 0" shape is the only form the solving formula in Step 3 understands.

Answer (b)x² + 6x − 90 = 0

STEP 3 · PART (c)

Solve with the quadratic formula

Why this formula: our equation won't split into nice whole-number brackets (nothing multiplies to −90 and adds to 6). When factoring fails, the quadratic formula always works on any equation shaped like ax² + bx + c = 0.

The raw formula — learn this shape

x = ( −b ± √(b² − 4ac) ) ÷ 2a

athe number in front of x² → here 1

bthe number in front of x → here 6

cthe number on its own → here −90

±means do it twice: once with + and once with −, giving two answers

↓ plug a = 1, b = 6, c = −90 in ↓

Now fill in the numbers x = ( −6 ± √(6² − 4·1·(−90)) ) ÷ 2 x = ( −6 ± √(36 + 360) ) ÷ 2 x = ( −6 ± √396 ) ÷ 2 = ( −6 ± 19.9 ) ÷ 2 → x = 6.9 or x = −12.9

Two possible answersx = 6.9 or x = −12.9

STEP 4 · FINISH

Pick the answer that makes sense

Why we throw one away: x is the width of a banknote — a real length. A length can never be negative, so −12.9 cm is impossible. Maths gives us both, but only one fits the real world.

So the width is x = 6.9 cm. Let's sanity-check it: a quick check builds confidence the answer is right.

Slide x to see the area area = 0.0 cm²

target 90

Width = 6.9 cm, length = 12.9 cm. Land the bar on the green line!

Check by multiplying back width × length = 6.9 × 12.9 → = 89.01 ≈ 90 cm² ✓ matches

Answer (c)Width ≈ 6.9 cm

🎉 ALL DONE

Quick recap

The whole trick: turn words into algebra, shape it into "= 0", then let the quadratic formula do the heavy lifting.

1Length in terms of xx + 6

2The equationx² + 6x − 90 = 0

3Quadratic formula gives6.9 or −12.9

4Width (must be positive)6.9 cm

Burn this into memory: the quadratic formula x = (−b ± √(b²−4ac)) ÷ 2a solves any equation shaped like ax² + bx + c = 0. Always rearrange to "= 0" first, and always sense-check which answer is real.