Word Problems and Applications

Let the smaller odd number be \(x\). The next consecutive odd number is therefore \(x+2\). Thus, \(x(x+2)=399\), giving \(x^2+2x-399=0\). Factoring, \((x-19)(x+21)=0\), so the positive value is \(x=19\), and the larger number is \(19+2=21\). Option 19 is the smaller number, not the larger one. Exam tip: consecutive odd numbers always differ by 2.

Let the smaller number be x. The next consecutive even number is x+2, so x(x+2)=528. This gives x²+2x−528=0, which factors as (x−22)(x+24)=0. Since the numbers are positive, x=22; hence the numbers are 22 and 24. Exam tip: consecutive even integers always differ by 2.

Let the breadth be x metres. Then the length is (x + 14) metres. Using area = length × breadth, we get x(x + 14) = 480, or x² + 14x − 480 = 0. Factoring gives (x − 16)(x + 30) = 0, so x = 16 or x = −30. Since a length cannot be negative, the breadth is 16 m. Exam tip: In rectangle word problems, first use area = length × breadth and reject any negative dimension.

Let the breadth of the board be \(x\) cm. Its length will then be \(x+15\) cm. Using the area, \(x(x+15)=250\), so \(x^2+15x-250=0\). Factoring gives \((x-10)(x+25)=0\), so \(x=10\) or \(x=-25\). Since a dimension cannot be negative, the breadth is 10 cm. Check: \(10\times25=250\) square cm. Exam tip: In rectangle word problems, assign \(x\) to one side first and form the area equation before solving.

Let the number be \(x\). Then \(x(x+12)=405\), which gives \(x^2+12x-405=0\). Factoring, \((x-15)(x+27)=0\), so \(x=15\) or \(x=-27\). Since the question asks for a positive number, the correct answer is 15. Exam tip: translate “12 greater” as \(x+12\) and check both roots before applying the stated condition.

Let the larger number be \(x\). Then the other number is \(x-7\), so \(x(x-7)=198\). Thus, \(x^2-7x-198=0\), which factors as \((x-18)(x+11)=0\). The roots are \(x=18\) and \(x=-11\). Since the numbers are positive, the valid larger number is 18, and the other number is 11. Exam tip: In such problems, represent the larger number by \(x\) and the number 7 less than it by \(x-7\).

Let the smaller number be (x) and the larger be (x+6), then (x(x+6)=160) gives (10) and (16). The larger number is (16).

The direct answer is option B, 17. Let the smaller number be x. Since the difference is 8, the larger number is x+8. Their product is 425, so x(x+8)=425, or x^2+8x-425=0. Factoring gives (x-17)(x+25)=0, so x=17 or x=-25. For the intended positive numbers, the smaller number is 17; the pair is 17 and 25, whose difference is 8 and product is 425. Option A, 15, would give the other number 23 and product 345, so it fails. Option B, 17, works exactly. Option C, 20, would give 28 and product 560, so it is wrong. Option D, 25, is actually the larger number in the correct pair, not the smaller one. Exam cue: when a difference is given, write the larger number as smaller number plus the difference, then use the product.

Let the breadth be \(x\) cm. Then the length is \(x+3\) cm. Using the area formula, \(x(x+3)=154\), so \(x^2+3x-154=0\). Factoring gives \((x-11)(x+14)=0\), yielding \(x=11\) or \(x=-14\). Since length and breadth must be positive, \(-14\) is not physically valid. Therefore, the breadth is 11 cm. Exam tip: In dimension-based word problems, reject any negative root.

If breadth is (x), then (x(x+5)=126), so (x=9) and length (14). Before answering, check the word asked.

If the side of the square is \(x\) cm, its area is \(x^2\). Thus, \(x^2=361\), so \(x=\sqrt{361}=19\). A side length cannot be negative, so \(-19\) is rejected. Exam tip: When the area of a square is given, find its side by taking the square root.

(x^2=400) gives side (20) m and perimeter (4x=80) m. Area and perimeter formulas are different.

The equation is (x(x+6)=187), giving (x=11). For total number, multiply rows and number per row.

(x(x+1)=272) gives (x=16). In an arrangement, total (=) number of groups (\times) objects per group.

The total-seat condition gives \(x(x-2)=224\). Thus, \(x^2-2x-224=0\), which factors as \((x-16)(x+14)=0\). Hence, \(x=16\) or \(x=-14\). Since the number of rows cannot be negative, the valid answer is 16. Exam tip: In word problems, always check the roots against the conditions of the real situation, such as being a positive integer.

The total number of flowers gives \(x(x+7)=330\), or \(x^2+7x-330=0\). Factoring gives \((x-15)(x+22)=0\), so \(x=15\) or \(x=-22\). Since the number of rows cannot be negative, the valid answer is 15. In word problems, always reject a root that is impossible in the given context.

The total number of chairs gives \(x(x+10)=416\), or \(x^2+10x-416=0\). Factoring, \((x-16)(x+26)=0\), so \(x=16\) or \(x=-26\). Since the number of rows cannot be negative, the valid answer is 16. In such word problems, always check whether the root is meaningful in context.

The total number of trees gives the equation \(x(x+4)=221\). Thus, \(x^2+4x-221=0\), which factors as \((x-13)(x+17)=0\). Hence, \(x=13\) or \(x=-17\), but the number of rows cannot be negative, so the correct answer is 13. Exam tip: In word problems, check the roots against the given context and reject impossible negative values.

The total is the number of days multiplied by the questions solved each day, so x(x + 3) = 130. This gives x² + 3x − 130 = 0 = (x + 13)(x − 10), whose roots are −13 and 10. Since a number of days cannot be negative, the student practised for 10 days.

The equation is (x(x+5)=216), which gives (x=12). In group-based questions, the total number is formed by multiplication.

Let the number be x. The statement says x² + 6 = 202, because 6 is added to the square of the number. Subtracting 6 from both sides gives x² = 196. Since 196 = 14², the possible values of x are 14 and −14. The question specifically asks for the positive value, so x = 14 and option C is correct. Option A would give 144 + 6 = 150, option B would give 169 + 6 = 175, and option D would give 225 + 6 = 231; none of these equals 202. The governing idea is translating a verbal statement into a quadratic equation and then selecting the root that satisfies the stated sign condition.

Let the number be x. “25 is subtracted from the square” translates to x² − 25 = 119. Adding 25 to both sides gives x² = 144. Since 144 = 12², x can be 12 or −12, but the question asks for the positive number. Therefore x = 12, so option B is correct. A quick check confirms the result: 12² − 25 = 144 − 25 = 119. Option A gives 100 − 25 = 75, option C gives 196 − 25 = 171, and option D gives 256 − 25 = 231. The important governing concept is careful translation of subtraction in a word problem, followed by taking the appropriate square-root value rather than automatically accepting both signs.

Let the number be \(x\). Then \(x^2+39=208\), so \(x^2=208-39=169=13^2\). Thus, \(x=13\) or \(-13\); since the question asks for the positive number, the correct answer is 13. Exam tip: isolate the squared term first, then take the square root and check the sign condition.

Taking breadth as (x), (x(x+18)=648), giving (x=18). In a rectangular region, make area the base equation.

Direct answer: The length is 46 m, so option D is correct. Let the breadth be x metres. The length is 21 m more, so it is x+21. Area equals length × breadth, hence x(x+21)=550. Expanding gives x^2+21x−550=0. Factoring: (x−25)(x+46)=0, so x=25 or x=−46. A length cannot be negative, so breadth = 25 m. Therefore length = 25+21 = 46 m. Option D is correct. Option A, 22 m, does not make the product 550 with a breadth 1 m; it is not the calculated length. Option B, 25 m, is the breadth, not the length. Option C, 40 m, does not produce the required dimensions and difference. The negative algebraic root is rejected because physical measurements are positive. Memory cue: first find the smaller dimension, then add the stated difference to obtain the length.