Question

Djamila is looking for conjugate pair solutions to the equation 3x2−5x=−6. Which of the following options should she choose?(1 point)%0D%0AResponses%0D%0A%0D%0A56±i47√6%0D%0AStart Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction%0D%0A%0D%0A−56±i47√6%0D%0Anegative Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction%0D%0A%0D%0A56±5i6%0D%0AStart Fraction 5 over 6 End Fraction plus or minus Start Fraction 5 i over 6 End Fraction%0D%0A%0D%0A56±7i6

Answer 1

To solve the equation $3x^2-5x=-6$, we first rearrange it to standard form:

$$3x^2-5x+6=0$$

Next, we can use the quadratic formula to find the roots of the equation $a{x}^2+bx+c=0$, which is given by:

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

In our case, $a=3$, $b=-5$, and $c=6$. Now we will calculate the discriminant:

$$b^2-4ac=(-5{)}^2-4\cdot 3\cdot 6=25-72=-47$$

Since the discriminant is negative ($-47$), we will have complex (conjugate pair) solutions. Now let's apply the quadratic formula:

$$x=\frac{-(-5)\pm \sqrt{-47}}{2\cdot 3}=\frac{5\pm \sqrt{-47}}{6}$$

This can be rewritten as:

$$x=\frac{5}{6}\pm \frac{i\sqrt{47}}{6}$$

Now we can match our results with the options provided. The conjugate pair solutions are:

$$\frac{5}{6}\pm \frac{i\sqrt{47}}{6}$$

Thus, the correct option that matches this solution format is:

Start Fraction 5 over 6 End Fraction plus or minus Start Fraction i Start Root 47 End Root over 6 End Fraction.