Question

find ( x ) and then ( overline{ab} ) and ( overline{bc} ).
note: solving a quadratic equation will result in two possible answers. always make sure the value of ( x ) produces positive lengths.
( x=)
( overline{ab}=) ( overline{bc}=)

Explanation:

Step1: Set up the equation

Since \(AB = BC\) (triangle \(ABC\) is isosceles), we have \(x^{2}+8x=6x + 15\).

Step2: Rearrange the equation

Move all terms to one side: \(x^{2}+8x-(6x + 15)=0\), which simplifies to \(x^{2}+2x-15=0\).

Step3: Factor the quadratic equation

Factor \(x^{2}+2x - 15\): \((x + 5)(x-3)=0\).

Step4: Solve for \(x\)

Set each factor equal to zero:

• \(x+5 = 0\) gives \(x=-5\)
• \(x - 3=0\) gives \(x = 3\)

Since length cannot be negative (when \(x=-5\), \(AB=(-5)^{2}+8\times(-5)=25 - 40=-15\) and \(BC=6\times(-5)+15=-30 + 15=-15\)), we discard \(x=-5\).

Step5: Calculate \(AB\) and \(BC\)

When \(x = 3\):

• \(AB=x^{2}+8x=3^{2}+8\times3=9 + 24=33\)
• \(BC=6x+15=6\times3+15=18 + 15=33\)

Answer:

\(x = 3\), \(\overline{AB}=33\), \(\overline{BC}=33\)