Answer:
(a) when x = 2: width = 2 units and length = 5 units
(b) 546 units²
Step-by-step explanation:
Given expression:
[tex]6x^2-5x-4[/tex]
Part (a)
To factor a quadratic in the form [tex]ax^2+bx+c[/tex]
Find 2 two numbers that multiply to [tex]ac[/tex] and sum to [tex]b[/tex]
Two numbers that multiply to -24 and sum to -5: -8 and 3
Rewrite [tex]b[/tex] as the sum of these 2 numbers:
[tex]\implies 6x^2+3x-8x-4[/tex]
Factorize the first two terms and the last two terms separately:
[tex]\implies 3x(2x+1)-4(2x+1)[/tex]
Factor out the common term [tex](2x+1)[/tex]:
[tex]\implies (3x-4)(2x+1)[/tex]
Therefore:
[tex]\implies x=\dfrac{4}{3}, \: x=-\dfrac{1}{2}[/tex]
As distance is positive, [tex]x > \dfrac{4}{3}[/tex]
Let [tex]x=2[/tex]
[tex]\implies (3(2)-4)(2(2)+1)[/tex]
[tex]\implies (2)(5)[/tex]
Therefore, possible dimensions of the rectangle are:
width = 2 units
length = 5 units
Part (b)
Substitute [tex]x=10[/tex] into the equation:
[tex]\begin{aligned}\implies 6(10)^2-5(10)-4&=600-50-4\\&=546\end{aligned}[/tex]
