What are the solutions to the quadratic equation 4x² + 22x + 36?

To solve the quadratic equation 4x² + 22x + 36 = 0, we can use the quadratic formula:

x = \frac{-b \pm \sqrt{b² – 4ac}}{2a}

Where a, b, and c are the coefficients from the equation in the form ax² + bx + c = 0. In our equation:

Now, let’s calculate the discriminant (b² – 4ac):

Now subtract:

b² – 4ac = 484 – 576 = -92

Since the discriminant is negative (-92), this means that the solutions will be complex (or imaginary) numbers. Let’s compute those:

Using the quadratic formula:

x = \frac{-22 \pm \sqrt{-92}}{2 * 4}

Now, we need to express the square root of a negative number:

\sqrt{-92} = \sqrt{92} * i = 2\sqrt{23} * i

Putting it all together:

x = \frac{-22 \pm 2\sqrt{23}i}{8}

Now, let’s simplify it:

x = \frac{-11 \pm \sqrt{23}i}{4}

Thus, the solutions to the quadratic equation 4x² + 22x + 36 = 0 are:

x = \frac{-11 + \sqrt{23}i}{4} \quad and \quad x = \frac{-11 – \sqrt{23}i}{4}

These solutions indicate that the roots of the equation are complex, meaning the graph of the quadratic does not intersect the x-axis.