Exercise
Given the equation:
For which values of parameter m, the line is below the x-axis.
Final Answer
Solution
Given the equation:
Open brackets:
Isolate x on one side:
We want to find out when the expression we got is less than zero, meaning
Solving inequality of a fraction is equivalent to solving an inequality of a multiplication, so instead of solving the inequality we got, we will solve the inequality:
Open brackets:
Reduce by 2:
It is a square inequality. its coefficients are:
The coefficient of the squared expression (a) is positive, so the parabola (quadratic equation graph) “smiles” (= bowl-shaped). The sign of the inequality means we are looking for the sections the parabola is below the x-axis. We find the solutions (= zeros = roots) of the quadratic equation using the quadratic formula. Putting the coefficients in the formula gives us
Hence, we get the solutions:
Since the parabola “smiles” and we are interested in the sections below the x-axis, we get
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