55. REASONING How many solutions does the equation ax2 + bx + c = 0 have when a and c have different
signs? Explain your reasoning.

From the picture, we can see that ΔLSP and ΔLRN are similar, so corresponding sides are proportionate:

LN : LP = 28:12 = 7:3

Therefore, the LRN sides is 7/3 of the corresponding side of LSP.

Then, it states that the area of LSP = 50, and area of a triangle is (1/2)bh, so we set up the equation

Area of LSP = (1/2)bh = 50 ← Remember how the corresponding sides are 7/3 of LSP? Therefore, the area of LRN:

LRN = (1/2)(7b/3)(7h/3) ← Take out the 7/3 and multiply them together

= (49/9)(1/2)bh ← From LSP, we know that (1/2)bh = 50, so plug that in

= (49/9)*50 ≈ 272.222 units ²