How do you find the zeros, real and imaginary, of #y=- 7x^2-15x -35# using the quadratic formula?

Question

How do you find the zeros, real and imaginary, of #y=- 7x^2-15x  -35# using the quadratic formula?

Kennedy Adams · Accepted Answer

2 imaginary roots:
#x = (-15 +- sqrt755)/14# #y = - (7x^2 + 15x + 35) = 0# Use new improved quadratic formula (Socratic Search) #D = d^2 = b^2 - 4ac = 225 - 980 = - 755 = 755i^2# --> #d = +- isqrt755# Because D < 0, there are 2 imaginary roots. #x = - b/(2a) +- d/(2a) = - 15/14 +- isqrt755/14 = (-15 +- isqrt755)/14# Note. Using the improved quadratic formula, the proceeding is much simpler, and the numeric computation is much easier.

Abigail Allen · Answer

#x=(15+isqrt(755))/(-14),##(15-isqrt(755))/(-14)#

The zeros of a quadratic equation are the values for #x#, and are where the parabola crosses the x-axis.

#y=-7x^2-15x-35#

Substitute a zero for the #y#.

#0=-7x^2-15x-35# is a quadratic equation in the form: #ax^2+bx+c=0#, where #a=-7#, #b=-15#, #c=-35#.

Use the quadratic formula to solve for #x# (the zeros).

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

Substitute the given values for #a, b, and c# into the formula.

#x=(-(-15)+-sqrt((-15)^2-4*-7*-35))/(2*-7)#

Simplify.

#x=(15+-sqrt(-755))/(-14)#

#x=(15+isqrt(755))/(-14),##(15-isqrt(755))/(-14)#

Julia Adams · Answer

undefined To find the zeros of the quadratic equation $y = -7x^2 - 15x - 35$ using the quadratic formula, you first identify the coefficients of $x^2$, $x$, and the constant term. In this equation, $a = -7$, $b = -15$, and $c = -35$. Then, substitute these values into the quadratic formula:

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

Substitute the values of $a$, $b$, and $c$:

$$x = \frac{{-(-15) \pm \sqrt{{(-15)^2 - 4(-7)(-35)}}}}{{2(-7)}}$$

Simplify the expression inside the square root:

$$x = \frac{{15 \pm \sqrt{{225 - 980}}}}{{-14}}$$

$$x = \frac{{15 \pm \sqrt{{-755}}}}{{-14}}$$

Since the discriminant $(b^2 - 4ac)$ is negative ($-755$), the solutions will be imaginary.

Thus, the zeros of the equation $y = -7x^2 - 15x - 35$ are:

$$x = \frac{{15 \pm \sqrt{{-755}}}}{{-14}}$$