Question

solve the following word problems: 1. the square of a number is 20 more than 8 times the number. find the number. 2. the length of a rectangular table is 8 more than the width. if the area is 180 m², find the length and the width.

Explanation:

Step1: Set up an equation for problem 1

Let the number be $x$. The equation is $x^{2}=8x + 20$.

Step2: Rearrange the equation

Rewrite it as a quadratic - equation in standard form: $x^{2}-8x - 20=0$.

Step3: Factor the quadratic equation

$(x - 10)(x+2)=0$.

Step4: Solve for $x$

Set each factor equal to zero: $x - 10 = 0$ gives $x = 10$; $x+2 = 0$ gives $x=-2$.

Step5: Set up an equation for problem 2

Let the width of the rectangle be $w$. Then the length $l=w + 8$. The area formula $A=l\times w$ gives $(w + 8)w=180$.

Step6: Expand and rearrange the equation

$w^{2}+8w-180 = 0$.

Step7: Factor the quadratic equation

$(w + 18)(w - 10)=0$.

Step8: Solve for $w$

Set each factor equal to zero: $w+18 = 0$ gives $w=-18$ (rejected since width cannot be negative), $w - 10=0$ gives $w = 10$.

Step9: Find the length

Since $l=w + 8$, when $w = 10$, $l=10 + 8=18$.

Answer:

1. The numbers are $10$ and $-2$.
2. The width is $10$ m and the length is $18$ m.