Determine whether the quadratic equation 6d

Question

2
=−6d−3 has any real solutions. If so, solve for them.

No real solutions

d=−
2
1
​
+
6
3
​

​
, −
2
1
​
−
6
3
​

​


d=−1

d=0, 1

Answer 1

To determine whether the quadratic equation \(6d^2 = -6d - 3\) has any real solutions, we can first rewrite it in the standard form of a quadratic equation \(ax^2 + bx + c = 0\).

1. Rearranging the equation: \[ 6d^2 + 6d + 3 = 0 \]

2. Now identify \(a\), \(b\), and \(c\):

   • \(a = 6\)
   • \(b = 6\)
   • \(c = 3\)

3. Next, we calculate the discriminant \(D\) using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = 6^2 - 4 \cdot 6 \cdot 3 = 36 - 72 = -36 \]

Since the discriminant \(D\) is negative (\(-36\)), this means the quadratic equation has no real solutions.

Thus, the conclusion is:

• No real solutions.