Check Real Solutions 72E221

1. **State the problem:** We want to determine how to check if all solutions of an equation are real numbers. 2. **General rule for polynomials:** For a polynomial equation, the nature of solutions depends on the discriminant (for quadratics) or other criteria for higher degrees. 3. **Quadratic equations:** For a quadratic equation $$ax^2 + bx + c = 0$$, the discriminant is $$\Delta = b^2 - 4ac$$. 4. **Check discriminant:** - If $$\Delta > 0$$, there are two distinct real solutions. - If $$\Delta = 0$$, there is one real repeated solution. - If $$\Delta < 0$$, the solutions are complex (not real). 5. **Higher degree polynomials:** - Use methods like factoring, graphing, or applying the Rational Root Theorem. - Alternatively, use the polynomial's derivative and sign changes to estimate real roots. 6. **Summary:** To ensure all solutions are real, check the discriminant for quadratics or use appropriate methods for higher degree polynomials to verify no complex roots exist.