How do you find the zeros of # y = 3/2x^2 + 3/2x +9/2 # using the quadratic formula?

Question

How do you find the zeros of # y =  3/2x^2 + 3/2x +9/2  # using the quadratic formula?

Avery Alexander · Accepted Answer

#x=(-1+-isqrt(11))/2#

Finding the zeroes of the function is the same as solving the following equation:

#3/2x^2+3/2x+9/2=0#

Because fractions are quite annoying to deal with, I will multiply both sides by #2 \/ 3# before we use the quadratic formula:

#2/3(3/2x^2+3/2x+9/2)=0*2/3#

#x^2+x+3=0#

Now we can use the quadratic formula, which says that if we have a quadratic equation in the form:

#ax^2+bx+c=0#

The solutions will be:

#x=(-b+-sqrt(b^2-4ac))/(2a)#

In this case, we get:

#x=(-1+-sqrt((-1)^2-4*3))/2#

#x=(-1+-sqrt(1-12))/2#

#x=(-1+-sqrt(-11))/2#

#x=(-1+-isqrt(11))/2#

Avery Alexander · Answer

undefined To find the zeros of $ y = \frac{3}{2}x^2 + \frac{3}{2}x + \frac{9}{2} $ using the quadratic formula:

1. Identify the coefficients a, b, and c in the quadratic equation $ y = ax^2 + bx + c $. In this case, $ a = \frac{3}{2} $, $ b = \frac{3}{2} $, and $ c = \frac{9}{2} $.

2. Substitute the values of a, b, and c into the quadratic formula: $ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} $.

3. Plug in the values for a, b, and c:

$$ x = \frac{{-\frac{3}{2} \pm \sqrt{{\left(\frac{3}{2}\right)^2 - 4\left(\frac{3}{2}\right)\left(\frac{9}{2}\right)}}}}{{2\left(\frac{3}{2}\right)}} $$

4. Simplify the expression inside the square root:

$$ x = \frac{{-\frac{3}{2} \pm \sqrt{{\frac{9}{4} - \frac{27}{2}}}}}{{\frac{3}{2}}} $$

$$ x = \frac{{-\frac{3}{2} \pm \sqrt{{\frac{9}{4} - \frac{54}{4}}}}}{{\frac{3}{2}}} $$

$$ x = \frac{{-\frac{3}{2} \pm \sqrt{{-\frac{45}{4}}}}}{{\frac{3}{2}}} $$

5. Simplify the expression inside the square root:

$$ x = \frac{{-\frac{3}{2} \pm \frac{\sqrt{45}}{2}i}}{{\frac{3}{2}}} $$

6. Multiply the numerator and denominator by 2 to get rid of the fraction:

$$ x = \frac{{-3 \pm \sqrt{45}i}}{{3}} $$

7. Thus, the zeros of the quadratic equation are $ x = \frac{{-3 + \sqrt{45}i}}{{3}} $ and $ x = \frac{{-3 - \sqrt{45}i}}{{3}} $.