Function and equation are two basic concepts in junior high school mathematics. Although they are different in form, they are essentially linked and closely related. Such as: unary quadratic equation and quadratic function. We know that the equation in the form of ax2+bx+c=0 is a quadratic equation, while the equation in the form of y= ax2+bx+c(a, B and C are constants, and a≠0) is a quadratic function. They are almost the same in form, the only difference is that the expression of quadratic equation in one variable is equal to 0, while the expression of quadratic function is equal to y, which makes the relationship between them particularly close, and many questions are based on this. Why is this happening? Mainly because when the variable y in the quadratic function takes 0, the quadratic function becomes a quadratic equation with one variable. It can be seen that many knowledge points in the equation can be applied to the function. Below, let's learn more about the specific applications between them. 1. The application relationship between the matching method to solve the equation and the quadratic function is that one of the four methods to solve the equation is the matching method. In quadratic function, we often need to transform the general form into a new form. This transformation process is actually to formulate it, just like the equation formula. Example 1: Solving the equation solution by matching method: (1) (2) (3) (4) ... Example 2: Pointing out the vertex coordinates of the function. Solution: (5)(6)(7)(8)∴ The vertex is four steps (1), (2), (3), (4) and functions (5) and (6) in the equation (-2,-17). It can be seen that the equation is closely related to the function. We know from the study of textbooks; When the image of the quadratic function y= ax2+bx+c(a≠0) intersects the X axis, the value of the abscissa of the intersection point is the root of the equation ax2+bx+c=0(a≠0). Second, the discriminant of the root of the quadratic equation of one variable is combined with the quadratic function. When the function has two intersections with the X axis, one intersection and no intersection respectively, the discriminant of the root of the quadratic equation of one variable corresponding to the function is: △ >; 0, delta = 0 and delta; 0, the equation has two unequal real roots; When △ = 0, the equation has two equal real roots; When △ 0, there are two intersections; If △ = 0, there is an intersection; if△； 0。 Analysis: The first method is to use the matching method to explain in the form of Y = (x-2) 2+ 1 (But if the coefficient value is not good, this method will be more troublesome. ) the second method: use delta to explain, because delta =-4; 0, so the image opening is upward. So the image is on the X axis, so no matter what value X takes, y >；; 0。 Example 5: Prove that no matter what real number M takes, the equation x2-(m2+m) x+m-2 = 0 must have two unequal real number roots. Analysis: if this problem is done in the conventional way, it is to prove the delta of a quadratic equation with one variable. The problem of 0. However, the discriminant △ of this problem is a univariate quartic polynomial about m, and the sign is difficult to judge, which brings trouble to the proof. If we analyze the meaning of the problem with the idea of function, let f (x) = x2-(m2+m) x+m-2, because its opening is upward, we only need to find a real number x0, so f (x0).