Solve 4d

Question

2
−4d=3 using the quadratic formula.

d=
2
3
​
, −
2
1
​


d=
2
1
​
+
2
2
​

​
,
2
1
​
−
2
2
​

​


d=
8
4+4
2
​

​
,
8
4−4
2
​

​


d=−2,
2
1
​

Answer 1

To solve the quadratic equation \( 4d^2 - 4d - 3 = 0 \) using the quadratic formula, we start with the formula:

\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, \( a = 4 \), \( b = -4 \), and \( c = -3 \).

1. Calculate \( b^2 - 4ac \):

\[ b^2 - 4ac = (-4)^2 - 4(4)(-3) = 16 + 48 = 64 \]

2. Apply the quadratic formula:

Now, substituting into the formula:

\[ d = \frac{-(-4) \pm \sqrt{64}}{2(4)} \] \[ d = \frac{4 \pm 8}{8} \]

3. Calculate the two possible values for \( d \):

• First value:

\[ d = \frac{4 + 8}{8} = \frac{12}{8} = \frac{3}{2} \]

• Second value:

\[ d = \frac{4 - 8}{8} = \frac{-4}{8} = -\frac{1}{2} \]

So, the solutions for the equation \( 4d^2 - 4d - 3 = 0 \) are:

\[ d = \frac{3}{2}, \quad d = -\frac{1}{2} \]

The final answer is:

\[ d = \frac{3}{2}, -\frac{1}{2} \]