Answer:
4,28
Step-by-step explanation:
So let's look at the sentence to take any necessary information out. It says the length of a rectangle plus it's width is 32. Since we do not know what the length and width are, let's just represent them as L (length) and W (width). Using this information we get the equation: [tex]L + W = 32[/tex] and since we're given the area of the rectangle, we can also define another equation which is: [tex]LW = 112[/tex] since the length * width = area.
So now all we have is a systems of equations. We can solve for L or W in the area equation and plug it into the addition equation. We can also solve for L or W in the addition equation and then plug it into the area equation, either should work, but in this example I'll solve for L in the addition equation:
Original Equation:
[tex]L + W = 32[/tex]
Subtract 32 from both sides
[tex]L = 32-W[/tex]
Now let's use this definition of L to plug into the second equation:
[tex]LW = 112 \implies (32-W)W = 112[/tex]
Distribute the W
[tex]-W^2+32W=112[/tex]
Now if you notice, we simply have a quadratic equation and we can subtract 112 from both sides and then use the quadratic formula: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Subtract 112 from both sides
[tex]-W^2+32W-112=0[/tex]
Use quadratic formula
[tex]W = \frac{-32\pm\sqrt{32^2-4(-1)(-112)}}{2(-1)}[/tex]
Simplify stuff in radical
[tex]W = \frac{-32\pm\sqrt{576}}{-2}[/tex]
Calculate the square root
[tex]W=\frac{-32\pm24}{-2}[/tex]
Use positive sign:
[tex]W=\frac{-32+24}{-2}[/tex]
Simplify numerator
[tex]W = \frac{-8}{-2}[/tex]
Simplify fraction
[tex]W=4[/tex]
Take negative sign
[tex]W = \frac{-32-24}{-2}[/tex]
Simplify numerator
[tex]W=\frac{-56}{-2}[/tex]
W = [tex]28[/tex]
So we have two solutions, and the reason for this is because had we solved for W in the original equation and substituted that into here, there would be no different, it's not bound by some variable name. So these are actually the two sides of the rectangles
4 + 28 = 32
4 * 28 = 112
4, 28
