The base of a triangular sign exceeds the height by 8 inches. If the area of the sign is 24 square inches, find the length of the base and the height of the triangle.

Answer:
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The height is: "4 inches" ;
and the base length is: "12 inches" .
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Explanation:
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The formula of the area of a triangle is:

Area = (1/2) * (base length) * (perpendicular height) ; or write as:

A = (1/2) * (b) * (h) ;
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Given: A = 24 in² ;
              b = h + 8 ;
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Find: "b" ; and: "h" ;
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→ Since: " A = (1/2) * b * h " ;

     → Plug in our known values:

            → 24 = (1/2) * (h + 8) * h ;
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Find: h ;
Find "b" , which is: "(h +8)" ;
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We have:

  → 24 = (1/2) * (h + 8) * h ;

Multiply EACH SIDE of the equation by "2" ; to get rid of the fraction:
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→ 2 * {24 = (1/2) * (h + 8) * h} ;

     → to get:
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     → 48 = 1 * (h + 8) * h ;

↔ Rewrite: h(h + 8) = 48 ;
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Note the "distributive property of multiplication":
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    a(b+c) = ab + ac ;

    a(b−c) = ab − ac ;
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→ So; h(h + 8) = h*h + h*8 = h² +8h = 48 ;
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We have: " h² + 8h = 48 " ; To solve for "h" ; let us see if we can
          write this equation in "quadratic format" ; that is:
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    " ax² + bx + c = 0 ; a ≠ 0 ; " ;
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We have:   h² + 8h = 48 ; Subtract "48" from EACH SIDE of the equation:
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                  h² + 8h − 48 = 48 − 48 ;
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to get:    h² + 8h − 48 = 0 ;
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Note that is equation IS, in fact, written in "quadratic format" ;
   that is: "ax² + bx + c = 0 ; a ≠ 0 " ;
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      in which: a = 1 ;
(Note: The "implied coefficient" of "1"; since anything multipled by "1" is that same result);
                      b = 8 ;
                      c = - 48;
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Now, let us see if we can solve by factoring; if we cannot, we can use the quadratic equation formula:
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Let us trying factoring: h² + 8h − 48 = (h+12) (h − 4) = 0 ;
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Since anything multiplied by "zero" equals "zero" ;

Then either:  (h+12) = 0 ; h = -12 ;
                     (h − 4) = 0 ; h = 4 ;
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So we have two (2) values for "h" ; "h = 4" , and "h = -12" .

So, which value do we use for "h"? Since "h" refer to "height";
we know that "height" cannot be a "negative value"; so we use:

"h = 4" .

Now, we are given: "b = h + 8 = 4 + 8 = 12 "
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So, h = 4 ; b = 12.
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Now check our work: "A = (1/2) (b) (h)" ; Given "A = 24" .

24 = (1/2) (12) (4)? 24 = (1/2) * 48 ? YES!
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So, the height is: "4 inches" ;
and the base length is: "12 inches" .
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