Word Problems and Applications

The equation is (x^2-6x=16), or (x^2-6x-16=0), giving (x=8). In difference questions, read the order carefully.

(x^2=7x+18) gives (x^2-7x-18=0), and the positive solution is (9). The phrase more than means addition.

(x^2+x=42) gives (x^2+x-42=0), so (x=6). Convert the sum of number and square directly into an equation.

(x^2+x=110) gives (x^2+x-110=0), whose positive solution is (10). While factoring, check factors of (110).

Consecutive odd numbers are (x) and (x+2), so (x(x+2)=143) gives (x=11). Consecutive odd numbers differ by (2).

Take consecutive even numbers as (x) and (x+2), then (x(x+2)=168) gives (x=12). Hence the larger number is (14).

(x(x+2)=195) gives (x=13), so the larger number is (15). For odd numbers, (x) and (x+2) is the correct form.

Let the smaller number be \(x\). The next consecutive even number is \(x+2\). Thus, \(x(x+2)=224\), giving \(x^2+2x-224=0\). Factoring gives \((x-14)(x+16)=0\), so \(x=14\) or \(x=-16\). Since the numbers are specified as positive, \(x=14\) is valid, and the numbers are 14 and 16. Exam tip: consecutive even numbers always differ by 2.

If breadth is (x), then (x(x+8)=240), giving (x=12). Check dimensions by (12 \times 20=240).

Let the breadth be \(x\) cm. Then the length is \(x+10\) cm, so the area condition gives \(x(x+10)=375\), or \(x^2+10x-375=0\). Factoring gives \((x-15)(x+25)=0\). Thus \(x=15\) or \(x=-25\); a dimension cannot be negative, so the breadth is 15 cm. The value 25 cm is the length, not the breadth. Exam tip: verify the result using \(15\times25=375\).

Taking breadth as (x), (x(x+9)=400), giving (x=16). Do not accept the wrong negative value as a dimension.

Let the number be \(x\). Then \(x(x+2)=63\), so \(x^2+2x-63=0\). Factoring gives \((x-7)(x+9)=0\), yielding \(x=7\) or \(x=-9\). Since the question asks for a positive number, the correct answer is 7. For comparison, choosing 9 gives \(9\times11=99\), not 63. Exam tip: In word problems, first represent the unknown number by \(x\) and express the related number in terms of \(x\).

Let the number be \(x\). The other number is \(x+4\), so \(x(x+4)=165\). Thus, \(x^2+4x-165=0\), which factors as \((x-11)(x+15)=0\). The roots are \(x=11\) and \(x=-15\), but the number is specified as positive; therefore, the correct answer is 11. Exam tip: “4 more than a number” means \(x+4\), not \(4x\).

If the larger number is (x), then (x(x-6)=112), giving (x=14). Use (x-6) for less than the number.

Let the smaller number be (x) and the larger be (x+5), then (x(x+5)=84) gives (7) and (12). The larger number is (12).

Putting the smaller number as (x) and larger as (x+3), (x(x+3)=108), giving (x=9). Write the difference in the correct direction.

Taking breadth as (x), (x(x+1)=30), so (x=5). Even with small numbers, forming the equation is important.

Let the breadth be \(x\) metres. Then the length is \(x+7\) metres. Using area, \(x(x+7)=260\), so \(x^2+7x-260=0\). Factoring gives \((x-13)(x+20)=0\), hence \(x=13\) or \(x=-20\). Since a length cannot be negative, the breadth is 13 metres. Exam tip: for a rectangle, always use area = length × breadth.

If the side is (x), then (x^2=64), so (x=8). In a square, area equals the square of the side.

(x^2=144) gives side (12) cm and perimeter (4x=48) cm. If perimeter is asked, do not stop after finding the side.