Zero Solutions A90Fc6

1. **State the problem:** We are given the equation $$h(-2x + 6) = -14(x - 2)$$ and asked to find the value of $$h$$ for which the equation has zero solutions. 2. **Rewrite the equation:** Distribute both sides: $$h(-2x + 6) = -14(x - 2)$$ $$-2hx + 6h = -14x + 28$$ 3. **Group like terms:** Bring all terms involving $$x$$ to one side and constants to the other: $$-2hx + 14x = 28 - 6h$$ 4. **Factor $$x$$ on the left side:** $$x(-2h + 14) = 28 - 6h$$ 5. **Analyze for zero solutions:** - If the coefficient of $$x$$ is zero, i.e., $$-2h + 14 = 0$$, then the equation becomes: $$0 imes x = 28 - 6h$$ - This simplifies to: $$0 = 28 - 6h$$ - For zero solutions, the equation must be a contradiction, meaning the right side is not zero when the left side is zero. 6. **Solve for $$h$$ when coefficient of $$x$$ is zero:** $$-2h + 14 = 0$$ $$-2h = -14$$ $$h = \frac{-14}{-2} = 7$$ 7. **Check the constant term when $$h=7$$:** $$28 - 6(7) = 28 - 42 = -14 \neq 0$$ 8. **Conclusion:** - When $$h = 7$$, the equation reduces to $$0 = -14$$, which is false. - This means there are zero solutions because the equation is a contradiction. **Final answer:** This equation will have zero solutions when $$h = 7$$ because you get zero solutions when you have zero number of $$x$$'s on either side of the equation and a nonzero constant term, resulting in a contradiction.