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Answer :
Let the length be [tex]$x$[/tex] (in cm) and the width be [tex]$(x-7)$[/tex] (in cm), since the length is 7 cm greater than the width. The area of the rectangle is given as 120 mm² (here, we assume consistent units for the dimensions). Thus, we have the equation
[tex]$$
x(x-7)=120.
$$[/tex]
Expanding the equation gives
[tex]$$
x^2 - 7x - 120 = 0.
$$[/tex]
This is a quadratic equation of the form
[tex]$$
ax^2 + bx + c = 0,
$$[/tex]
with [tex]$a=1$[/tex], [tex]$b=-7$[/tex], and [tex]$c=-120$[/tex]. The discriminant [tex]$D$[/tex] is calculated as
[tex]$$
D = b^2 - 4ac = (-7)^2 - 4(1)(-120) = 49 + 480 = 529.
$$[/tex]
Taking the square root of the discriminant, we get
[tex]$$
\sqrt{D} = \sqrt{529} = 23.
$$[/tex]
Now, applying the quadratic formula
[tex]$$
x = \frac{-b \pm \sqrt{D}}{2a},
$$[/tex]
we find the solutions:
[tex]$$
x_1 = \frac{7 + 23}{2} = \frac{30}{2} = 15 \quad \text{and} \quad x_2 = \frac{7 - 23}{2} = \frac{-16}{2} = -8.
$$[/tex]
Since a length cannot be negative, we discard [tex]$x_2 = -8$[/tex]. Thus, the length of the rectangle is
[tex]$$
\text{Length} = 15 \text{ cm},
$$[/tex]
and the width is
[tex]$$
\text{Width} = x - 7 = 15 - 7 = 8 \text{ cm}.
$$[/tex]
In summary, the rectangle's dimensions are 15 cm (length) and 8 cm (width), with a discriminant of 529 and [tex]$\sqrt{D} = 23$[/tex].
[tex]$$
x(x-7)=120.
$$[/tex]
Expanding the equation gives
[tex]$$
x^2 - 7x - 120 = 0.
$$[/tex]
This is a quadratic equation of the form
[tex]$$
ax^2 + bx + c = 0,
$$[/tex]
with [tex]$a=1$[/tex], [tex]$b=-7$[/tex], and [tex]$c=-120$[/tex]. The discriminant [tex]$D$[/tex] is calculated as
[tex]$$
D = b^2 - 4ac = (-7)^2 - 4(1)(-120) = 49 + 480 = 529.
$$[/tex]
Taking the square root of the discriminant, we get
[tex]$$
\sqrt{D} = \sqrt{529} = 23.
$$[/tex]
Now, applying the quadratic formula
[tex]$$
x = \frac{-b \pm \sqrt{D}}{2a},
$$[/tex]
we find the solutions:
[tex]$$
x_1 = \frac{7 + 23}{2} = \frac{30}{2} = 15 \quad \text{and} \quad x_2 = \frac{7 - 23}{2} = \frac{-16}{2} = -8.
$$[/tex]
Since a length cannot be negative, we discard [tex]$x_2 = -8$[/tex]. Thus, the length of the rectangle is
[tex]$$
\text{Length} = 15 \text{ cm},
$$[/tex]
and the width is
[tex]$$
\text{Width} = x - 7 = 15 - 7 = 8 \text{ cm}.
$$[/tex]
In summary, the rectangle's dimensions are 15 cm (length) and 8 cm (width), with a discriminant of 529 and [tex]$\sqrt{D} = 23$[/tex].
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