Word Problems and Applications

The equation is (x^2-12x=589), or (x^2-12x-589=0), giving (x=31). In a difference statement, keep the subtraction order correct.

Let the number be (x\u0008). Then (x^2+x=342\u0008), or (x^2+x-342=0\u0008). Factoring gives ((x-18)(x+19)=0\u0008), so (x=18\u0008) or (x=-19\u0008). Since the question asks for the positive number, the answer is 18. Exam tip: Form the equation (x^2+x\u0008) first and then select the root that satisfies the stated condition.

(x^2+x=506) gives (x^2+x-506=0), whose positive solution is (22). While factoring, check factors of (506).

Let the original number be \(x\). Then \((x+9)^2=676\). Taking the positive square root gives \(x+9=26\), so \(x=26-9=17\). Option 26 is the resulting number, not the original number. In exams, remember that a square equation generally has positive and negative roots, but here the question specifically asks for the positive root.

Let the original number be \(x\). Then \((x-11)^2=196\), so \(x-11=\pm14\). This gives \(x=25\) or \(x=-3\). Since the original number is required to be positive, \(x=25\) is correct. In exams, remember to consider both signs when taking the square root and then apply the given condition.

By Pythagoras, (x^2+(x+12)^2=60^2), giving (x=36). Always take the hypotenuse as the largest side.

By the Pythagorean theorem, \(x^2+(x+9)^2=45^2\). Simplifying gives \(x^2+9x-972=0\), which factors as \((x-27)(x+36)=0\). Thus, \(x=27\) or \(x=-36\). Since a length cannot be negative, the shorter leg is 27 m. Exam tip: Reject any negative root when the variable represents a length.

If breadth is (x) and length is (x+7), then (2(x+x+7)=74), giving (x=15). In perimeter questions, use (2(l+b)).

If breadth is (x), then (2(x+x+9)=90), giving (x=18) and length (27). Choose the dimension asked carefully.

The equation is (x(x+8)=425), giving (x=17). For total number, multiply rows and number per row.

The total number of books equals the number of racks multiplied by the books in each rack, so \(x(x+6)=391\). Thus, \(x^2+6x-391=0\), which factors as \((x-17)(x+23)=0\). The roots are \(x=17\) and \(x=-23\); since the number of racks cannot be negative, \(x=17\) is correct. Exam tip: In real-life word problems, reject a negative root when the variable represents a quantity.

The total-seat condition gives x(x + 5) = 500. Thus, x² + 5x − 500 = 0 = (x − 20)(x + 25), so x = 20 or x = −25. Since the number of rows cannot be negative, the valid answer is 20 rows. In such word problems, reject roots that are not meaningful in the given context.

The total number of plants gives \(x(x+11)=702\). Thus, \(x^2+11x-702=0\), which factors as \((x-26)(x+27)=0\). So \(x=26\) or \(x=-27\); a number of rows cannot be negative, so \(x=26\) is correct. Remember that 27 is the number of plants in each row, not the number of rows.

(x(x+12)=640) gives (x=20). For total objects, multiply groups and objects per group.

The total number of questions gives the equation x(x + 7) = 408, or x² + 7x − 408 = 0. Factoring gives (x − 17)(x + 24) = 0, so x = 17 or x = −24. Since the number of days cannot be negative, x = 17 is the valid answer. As an exam tip, verify it by checking that 17 × 24 = 408.

The equation is (x(x+9)=700), giving (x=20). Multiply the number of groups by objects per group.

Let the number be \(x\). Then \(x^2+71=512\), so \(x^2=512-71=441=21^2\). Hence \(x=\pm21\), and the positive number is \(21\). Option 20 is incorrect because \(20^2+71=471\), not 512. Exam tip: Taking a square root gives both positive and negative roots, but choose the positive root when the question asks for a positive number.

Let the number be \(x\). Then \(x^2-119=457\), so \(x^2=576=24^2\). Thus \(x=\pm24\), but the positive number is \(24\). In the exam, use the word “positive” to reject the negative root.

The square area is (18^2=324), and (12x=324) gives (x=27). For equal areas, equate both areas.

The area of a square is
x^2
, so
x^2=729
. Since
729=27^2
and a side length must be positive,
x=27
metres. The negative root,
-27
, is not valid for a length. Exam tip: When the area of a square is given, find its side by taking the positive square root.