Word Problems and Applications

Total work is (x(x+9)=736), giving (x=23). For total work, multiply days and work per day.

The total number of questions gives the equation \(x(x+11)=900\), or \(x^2+11x-900=0\). Factoring gives \((x-25)(x+36)=0\), so \(x=25\) or \(x=-36\). Since the number of days cannot be negative, \(x=25\) is correct. In such word problems, always reject roots that are not meaningful in the given context.

The total number of books gives the equation \(x(x+13)=1014\). Thus, \(x^2+13x-1014=0\), which factors as \((x-26)(x+39)=0\). Hence, \(x=26\) or \(x=-39\). Since the number of shelves cannot be negative, \(x=26\) is correct. The value 39 represents the number of books per shelf when \(x=26\), not the number of shelves. Exam tip: In word problems, reject any algebraic root that is not meaningful in the given context.

The total-chair condition gives x(x+17)=1568. Thus, x²+17x−1568=0, which factors as (x−32)(x+49)=0. Hence x=32 or x=−49. Since the number of rows cannot be negative, x=32 is the valid answer. Exam tip: In arrangement problems, reject negative roots and retain the positive integer root.

The total number of plants gives the equation \(x(x-9)=850\). Thus, \(x^2-9x-850=0\), which factors as \((x-34)(x+25)=0\). Therefore, \(x=34\) or \(x=-25\). Since the number of rows cannot be negative, the valid answer is 34. Check: \(34\times(34-9)=34\times25=850\). Exam tip: Substitute the answer back into the original situation and reject any negative value that is not meaningful in context.

The total number of questions gives \(x(x-7)=800\). Thus, \(x^2-7x-800=0\), which factors as \((x-32)(x+25)=0\). Hence, \(x=32\) or \(x=-25\). Since the number of students cannot be negative, the valid answer is \(32\). Exam tip: In such word problems, first express the total as ‘number of groups × items per group’, then reject any root that is not meaningful in the given context.

Let the original number be \(x\). The required equation is \((x+4)(x+10)=952\). On expanding, \(x^2+14x-912=0\), which factors as \((x-24)(x+38)=0\). Thus, \(x=24\) or \(x=-38\); since the number is specified as positive, the correct answer is 24. Verification: \((24+4)(24+10)=28\times34=952\). In an exam, substitute the chosen value back into the original product to verify it.

Let the original number be \(x\). Then \((x-7)(x+12)=756\), which gives \(x^2+5x-840=0\). Applying the quadratic formula yields \(x=\frac{-5\pm\sqrt{3385}}{2}\). The positive value is \(\frac{-5+\sqrt{3385}}{2}\); the other root is negative. Exam tip: translate ‘7 less’ as \(x-7\) and ‘12 more’ as \(x+12\) before forming the equation.

Let the positive number be \(x\). Then \((x-5)(x+14)=1242\). Substituting \(x=32\), we get \((32-5)(32+14)=27\times46=1242\), so the original number is 32. Exam tip: In such word problems, testing the options in the equation can be a quick alternative to solving the quadratic equation completely.

Let the number be \(x\). Then \(x^2=18x+432\), so \(x^2-18x-432=0\). Using the quadratic formula, \(x=\frac{18\pm\sqrt{18^2+4\cdot432}}{2}=9\pm3\sqrt{57}\). Since \(9-3\sqrt{57}\) is negative, the positive number is \(9+3\sqrt{57}\). Exam tip: form the quadratic equation first, calculate both roots, and then select the root that satisfies the condition in the question.

Let the number be \(x\). Then \(x^2-20x=621\), so \(x^2-20x-621=0\). Using the quadratic formula, \(x=\frac{20\pm\sqrt{400+2484}}{2}=10\pm\sqrt{721}\). Since the number is positive and \(10-\sqrt{721}<0\), the required value is \(10+\sqrt{721}\). Exam tip: translate the word statement into a quadratic equation first, then use the given condition, such as positivity, to select the valid root.

Let the number be \(x\). Then \(x^2+x=1260\), so \(x^2+x-1260=0\). Factoring gives \((x-35)(x+36)=0\), hence \(x=35\) or \(x=-36\). The positive number is \(35\). Exam tip: quickly recognise that \(1260=35\times36\), the product of two consecutive integers; 36 is incorrect because \(36^2+36=1332\).

(x^2+175=1076) gives (x^2=901), not a perfect square. If the data are inconsistent, do not choose options blindly.

Let the number be \(x\). According to the statement, \(x^2-256=1040\), so \(x^2=1296=36^2\). Thus, \(x=36\) or \(x=-36\), but the positive number is \(36\). For instance, choosing 34 gives \(34^2-256=900\), so it is not correct. Exam tip: Interpret “subtracted from the square” as \(x^2-\text{given value}\).

((x+9)^2=1681) and (x+9=41), so (x=32). If the positive case is clear, take the positive square root.

((x-11)^2=1156) and (x-11=34), so (x=45). After taking square root, choose the sign according to the condition.

Let the son's age be \(x\) years. Then the father's age is \(x+32\) years, so \(x(x+32)=1280\). This gives \(x^2+32x-1280=0\), or \((x-32)(x+64)=0\). Since age cannot be negative, \(x=32\), and the father's age is \(32+32=64\) years. Exam tip: In age problems, reject any negative root before selecting the answer.

If the daughter's age is (x), then (x(x+30)=1476), giving (x=24). In age problems, ignore the negative root.

The equation is \((x+7)(x-7)=1551\). Using the difference of squares, \(x^2-49=1551\), so \(x^2=1600\) and \(x=\pm 40\). Since age cannot be negative, the present age is 40 years. Exam tip: apply \((x+a)(x-a)=x^2-a^2\) directly in such age problems.

((x+6)(x-6)=1333) gives (x^2-36=1333) and (x=37). The difference of squares idea helps in such questions.