Question

is the solution shown below correct? explain.
9x+2=8x²+6x
-8x² + 3x + 2 = 0
x = -3 ± √((3)² - (4)(-8)(2))/(-16)
x = -3 ± √(9 - (64))/(-16)
x = (3 ± √(55)i)/16

Explanation:

Step1: Check Equation Rearrangement

Original equation: \(9x + 2 = 8x^2 + 6x\). Subtract \(8x^2 + 6x\) from both sides: \( -8x^2 + 3x + 2 = 0\). This step is correct.

Step2: Identify Quadratic Formula

For \(ax^2 + bx + c = 0\), quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a=-8\), \(b = 3\), \(c = 2\). So denominator should be \(2a=2\times(-8)=-16\), numerator's \(-b=-3\). This part is correct so far.

Step3: Calculate Discriminant

Discriminant: \(b^2 - 4ac=(3)^2 - 4\times(-8)\times(2)=9 + 64 = 73\) (not \(9 - 64\)). The error is here: \(4\times(-8)\times(2)=-64\), so \(-4ac=-4\times(-8)\times(2)=64\), not \(-64\). So discriminant calculation is wrong.

Step4: Check Subsequent Steps

Since discriminant was miscalculated, the steps after (involving imaginary numbers) are also wrong because discriminant \(73>0\), so roots are real, not complex. Also, when factoring out the negative sign later, the sign handling has errors too.

Answer:

The solution is incorrect. Errors include miscalculating the discriminant (\(b^2 - 4ac = 9 + 64 = 73\), not \(9 - 64\)) and incorrect handling of signs when simplifying, leading to wrong conclusions about complex roots (discriminant is positive, roots are real).