Word Problems and Applications

Let the smaller integer be (x), then (x(x+1)=72) gives (x=8). In exams, write consecutive numbers as (x) and (x+1).

Let the numbers be (x) and (x+1), then (x(x+1)=132) gives (x=11). So the larger integer is (12).

Taking breadth as (x), (x(x+3)=70), which gives (x=7). In area questions, do not forget length times breadth.

Area (=) length (\times) breadth, so (6x=96) is correct. In every word problem, first identify the formula.

If the side of the square is (x), then (x^2=169), so (x=13). A side length is always taken positive.

From (x^2=225), the side is (15) m and perimeter is (4x=60) m. First find the side, then apply perimeter.

If the number is (x), then (x(x+1)=56), giving (x=7). In the question, next integer means (x+1).

The equation is (x^2=5x+24), or (x^2-5x-24=0), whose positive solution is (x=8). If a positive number is asked, ignore the negative root.

(x^2=4x+45) gives (x^2-4x-45=0), and the positive solution is (9). In such questions, convert the sentence directly into an equation.

Take the numbers as (x) and (17-x), then (x(17-x)=72) gives (8) and (9). The smaller number is (8).

Putting the numbers as (x) and (19-x), (x(19-x)=90). The solutions are (9) and (10), so the larger number is (10).

If breadth is (x), then (x(x+5)=84), giving (x=7) and length (12). Check length and breadth separately at the end.

Taking breadth as (x), (x(x+4)=96), giving (x=8). In area questions, always note the unit.

If breadth is (x), then (x(x+2)=120), giving (x=10). Only the positive value is valid for a dimension.

(x(x+5)=150) gives breadth (10) and length (15). Give the answer according to the dimension asked.

By Pythagoras, (x^2+(x+7)^2=13^2), giving (x=5). In a right triangle, always take the hypotenuse as the largest side.

By the Pythagorean theorem, \(x^2+(x+1)^2=25\). This gives \(2x^2+2x-24=0\), or \(x^2+x-12=0\), which factors as \((x-3)(x+4)=0\). Since a side length cannot be negative, \(x=3\) cm, so the shorter side is 3 cm. The value 4 cm is the other perpendicular side, not the shorter one. Exam tip: verify the result using the familiar \(3,4,5\) right-triangle relationship.

By the Pythagorean theorem, \(x^2+(x+2)^2=10^2\). Simplifying gives \(2x^2+4x-96=0\), or \(x^2+2x-48=0\). Thus, \((x+8)(x-6)=0\), so \(x=-8\) or \(x=6\). Since a side length must be positive, the correct value is \(x=6\). In length-based problems, always reject a negative root.

Let the number be \(x\). Then \((x+3)^2=100\). Taking the positive square root gives \(x+3=10\), so \(x=7\). The negative square root would give \(x=-13\), but the question specifically asks for the positive square root. Exam tip: When solving a squared equation, consider both roots and then apply the condition given in the question.

Let the original number be \(x\). Then \((x-4)^2=81\), so \(x-4=9\) or \(x-4=-9\). Hence, \(x=13\) or \(x=-5\). Since the original number is positive, the correct answer is 13. Exam tip: always check both possible signs when solving a squared equation.