What are the solutions to the equation 4 x squared plus 3 x plus 1 equals 0?

Question

Answer options with 4 options
A.
StartFraction negative 3 plus i StartRoot 7 EndRoot over 8 EndFraction commma StartFraction negative 3 minus i StartRoot 7 EndRoot over 8 EndFraction

B.
StartFraction 3 plus i StartRoot 7 EndRoot over 8 EndFraction commma StartFraction 3 minus i StartRoot 7 EndRoot over 8 EndFraction

C.
StartFraction negative 3 plus 7 i over 8 EndFraction comma StartFraction negative 3 minus 7 i over 8 EndFraction

D.
StartFraction 3 plus 7 i over 8 EndFraction comma StartFraction 3 minus 7 i over 8 EndFraction

Answer 1

To solve the equation $4x^2+3x+1=0$, we will use the quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Here, $a=4$, $b=3$, and $c=1$.

First, we calculate the discriminant $b^2-4ac$:

$$b^2=3^2=9$$ $$4ac=4\cdot 4\cdot 1=16$$ $$b^2-4ac=9-16=-7$$

Since the discriminant is negative, the solutions will be complex numbers.

Now we substitute $a$, $b$, and the discriminant back into the quadratic formula:

$$x=\frac{-3\pm \sqrt{-7}}{2\cdot 4}$$

The square root of $-7$ can be expressed as $i\sqrt{7}$:

$$x=\frac{-3\pm i\sqrt{7}}{8}$$

Now we can express the two complex roots:

1. $x_1=\frac{-3+i\sqrt{7}}{8}$
2. $x_2=\frac{-3-i\sqrt{7}}{8}$

Thus, the solutions to the equation $4x^2+3x+1=0$ are:

A. $\frac{-3+i\sqrt{7}}{8},\frac{-3-i\sqrt{7}}{8}$

So, the correct answer is A.