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在中学代数中,如何确定方程实根的个数,通常要比解方程问题更灵活、更深刻、更带有启发性。因此在教学中,注意穿插这方面的知识和技能,对加强双基,提高学生的观察分析和解题能力是十分有益的。本文通过举例说明求方程实根个数有关方法。一、直接利用判别式对于实系数一元二次方程,可直接应用判别武确定实根的个数。例1 已知实数a、b、c满足a>0,b>a+c,试求方程ax~2+bx+c=0的实根的个数(参见第十一届全俄数学奥林匹克第三轮试题)。
In middle school algebra, how to determine the number of real roots of an equation is usually more flexible, deeper, and more suggestive than solving equation problems. Therefore, in teaching, attention should be paid to the knowledge and skills interspersed with this aspect, and it is very useful to strengthen the double-base and improve students’ ability to observe, analyze, and solve problems. This article illustrates how to find the number of real roots of the equation. First, the direct use of discriminant for the real coefficient of a quadratic equation can be directly applied to determine the number of real roots. Example 1 Knowing that real numbers a, b, and c satisfy a>0 and b>a+c, find the number of real roots of the equation ax~2+bx+c=0 (see the 11th All-Russian Mathematical Olympiad Three round questions).