if the roots of the Q.E. are equal then prove that b²=4ac.10 class quadratic equations.

If the roots of the equation are
equal:

x1 = x2,

that means
that the discriminant of the quadratic equation is equal to
zero.

delta = 0

delta = b^2 -
4ac

b^2 - 4ac = 0

We'll add
4ac both sides:

b^2 =
4ac

Let's see how to find delta. We'll write again the
q.e.:

ax^2 + bx + c = 0

We'll
factorize:

a(x^2 + bx/a + c/a) =
0

a(x^2 + 2bx/2a + b^2/4a^2 - b^2/4a^2+ c/a) =
0

We notice that we've modified the
ratio:

 bx/a = 2bx/2a

We've
also added and subtracted the quantity b^2/4a^2.

We've
completed the square x^2 + 2bx/2a + b^2/4a^2.

x^2 + 2bx/2a
+ b^2/4a^2 = (x + b/2a)^2

a[(x + b/2a)^2 - (b^2/4a^2
- c/a)] = 0

b^2/4a^2 - c/a = (b^2 -
4ac)/4a^2

b^2 - 4ac =
delta

a[(x + b/2a)^2 - (delta)/4a^2] =
0

(x + b/2a)^2 - (delta)/4a^2 =
0

(x + b/2a)^2 =
(delta)/4a^2

x + b/2a = sqrt
delta/2a

x1 = (-b+sqrt
delta)/2a

x2 = (-b-sqrt
delta)/2a

When x1 =
x2:

 (-b+sqrt delta)/2a = (-b-sqrt
delta)/2a

-b+sqrt delta = -b-sqrt
delta

We'll eliminate like
terms:

2sqrt delta = 0

delta =
0

b^2 - 4ac =
0

b^2 =
4ac