- 6x + 9 = 0.

Explanation:

If the roots of a quadratic equation are real and equal, then we know that the discriminant (b^2 - 4ac) must be equal to zero.

In the general form of a quadratic equation, ax^2 + bx + c = 0, the discriminant is b^2 - 4ac.

Setting the discriminant to zero, we get:

b^2 - 4ac = 0

b^2 = 4ac

Since we want the roots to be real and equal, we know that they must be -b/2a.

Substituting -b/2a for x in the quadratic equation, we get:

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

Simplifying, we get:

b^2 - 4ac = 0

b^2 = 4ac

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

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

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

b^2 - b^2/2 + c = 0

b^2/2 + c = 0

c = -b^2/2

Substituting -b^2/2 for c in the general form of a quadratic equation, we get:

ax^2 + bx - b^2/2 = 0

Multiplying by 2 to eliminate the fraction, we get:

2ax^2 + 2bx - b^2 = 0

Dividing by 2a to get the standard form of a quadratic equation, we get:

x^2 + bx/a - b^2/2a = 0

Simplifying, we get:

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

We can check that this is correct by using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Since the discriminant is zero, we get:

x = (-b ± sqrt(0)) / 2a

x = -b/2a

So the roots are real and equal.

Therefore, the quadratic equation whose roots are real and equal is:

x^2 + bx/a - b^2/2a = 0

or

2ax^2 + 2bx - b^2 = 0

which simplifies to:

x^2 + bx - b^2/2a = 0

For example, if a = 1, b = -6, and c = 9, then:

b^2 - 4ac = (-6)^2 - 4(1)(9) = 0

So the roots are real and equal.

The quadratic equation is:

x^2 - 6x + 9 = 0

which can be factored as:

(x - 3)^2 = 0

So the roots are x = 3 and x = 3.

Therefore,