Solving a mathematical problem about quadratic function Given that the image of quadratic function y = x square + ax + A-2 has two focal points with X axis, and the distance between the two focal points is 2 root sign 5, then a =? Hope to write out the answers and solutions

Solving a mathematical problem about quadratic function Given that the image of quadratic function y = x square + ax + A-2 has two focal points with X axis, and the distance between the two focal points is 2 root sign 5, then a =? Hope to write out the answers and solutions

Let y = x & sup2; + ax + A-2 have two intersections x1, X2 with x-axis,
And X1 < X2,
If x2-x1 = 2 √ 5 (1), then (x2-x1) & sup2; = 20,
X1 + x2 = - A (2) gives (x1 + x2) & sup2; = A & sup2;
If x1 · x2 = A-2 (3), 4x1x2 = 4a-8,
We get: (x1-x2) & sup2; = X1 & sup2; - 2x1x2 + x2 & sup2;
=x1²+2x1x2+x2²-4x1x2,
=（x1²+x2）²-4x1x2,
∴20=a²-4a+8,
That is a & sup2; - 4a-12 = 0,
（a-6）（a+2）=0,
∴a1=6,a2=-2.