Solve the equation .
Solve the equation(The complex numbers system) by following steps:
step0: Solve using the quadratic formula in the complex numbers system:
step1: Solve using the quadratic formula:
step2: Simplify the expression:
step3: Simplify the expression:
step4: Simplify the expression:
step5: Separate into possible cases:
step6: Simplify the expression:
step7: Simplify the expression:
step8: Rewrite:
step9: Remove the complex number(s):
The equation does not have any real solutions. This means that there are no real values of that satisfy the equation.
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The equation is a quadratic equation, which can be solved using the quadratic formula: . In this case, , , and . Plugging these values into the formula, you’ll find the values of where the function equals zero, which represent the points where the graph of the quadratic crosses the x-axis.
Quadratic equations like this one have their roots either at two distinct points, one point (if the discriminant is zero), or not at all (if the discriminant is negative). For our equation, first, we need to calculate the discriminant: . Since the discriminant is negative, you’ll find that this equation has no real solutions, indicating that its graph does not intersect the x-axis at any point. Instead, it lies entirely above the x-axis!