Question

② \\(\frac{x + 4}{2} = \frac{4x}{x - 6}\\)\\(\\(x + 4)(x - 6) = 8x\\)\\(x^2 - 6x + 4x - 24 = 8x\\)\\(x^2 - 2x - 24 = 8x\\)\\(\underline{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -8x\\ \\ \\ \\ \\ +8x\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\)\\(x^2 - 2x - 32x = 0\\)

Explanation:

Step1: Identify the equation type

The given equation is a rational equation: \(\frac{x + 4}{2}=\frac{4x}{x - 6}\). To solve it, we can use cross - multiplication (since if \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\) for \(b
eq0\) and \(d
eq0\)).

Step2: Cross - multiply

Cross - multiplying the equation \(\frac{x + 4}{2}=\frac{4x}{x - 6}\) gives \((x + 4)(x - 6)=2\times4x = 8x\).

Step3: Expand the left - hand side

Expand \((x + 4)(x - 6)\) using the FOIL method. \((x+4)(x - 6)=x\times x+x\times(-6)+4\times x + 4\times(-6)=x^{2}-6x + 4x-24=x^{2}-2x - 24\). So the equation becomes \(x^{2}-2x - 24 = 8x\).

Step4: Move all terms to one side

Subtract \(8x\) from both sides of the equation \(x^{2}-2x - 24 = 8x\) to get \(x^{2}-2x-8x - 24=0\), which simplifies to \(x^{2}-10x - 24 = 0\) (Note: There was a mistake in the original calculation when combining like terms. The correct combination of \(-2x-8x\) is \(-10x\), not \(-32x\)).

Step5: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b=-10\), \(c=-24\)), we can use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or factor the equation. Let's try factoring. We need two numbers that multiply to \(a\times c=1\times(-24)=-24\) and add up to \(b = - 10\). The numbers are \(-12\) and \(2\). So \(x^{2}-10x - 24=(x - 12)(x+2)=0\).
Setting each factor equal to zero:

• If \(x - 12=0\), then \(x = 12\).
• If \(x + 2=0\), then \(x=-2\).

We also need to check for extraneous solutions because in the original rational equation, the denominators \(2\) (which is never zero) and \(x - 6\) (so \(x
eq6\)). Both \(x = 12\) and \(x=-2\) are not equal to \(6\), so they are valid solutions.

Answer:

The solutions of the equation \(\frac{x + 4}{2}=\frac{4x}{x - 6}\) are \(x = 12\) and \(x=-2\) (after correcting the error in the original calculation's like - term combination).