How do you solve #-16x ^ { 2} = 12x ^ { 2} + 24x + 5#?

Question

Avery Alexander · Accepted Answer

See a solution process below:

First, add #color(red)(16x^2)# to each side of the equation to put the equation in standard form: #-16x^2 + color(red)(16x^2) = 12x^2 + color(red)(16x^2) + 24x + 5# #0 = (12 + color(red)(16))x^2 + 24x + 5# # 0 = 28x^2 + 24x + 5# #28x^2 + 24x + 5 = 0# Next. we can factor the left side of the equation as: #(14x + 5)(2x + 1) = 0# Now, we can solve each term on the left for #0#: Solution 1: #14x + 5 = 0# #14x + 5 - color(red)(5) = 0 - color(red)(5)# #14x + 0 = -5# #14x = -5# #(14x)/color(red)(14) = -5/color(red)(14)# #(color(red)(cancel(color(black)(14)))x)/cancel(color(red)(14)) = -5/14# #x = -5/14# Solution 2: #2x + 1 = 0# #2x + 1 - color(red)(1) = 0 - color(red)(1)# #2x + 0 = -1# #2x = -1# #(2x)/color(red)(2) = -1/color(red)(2)# #(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) = -1/2# #x = -1/2# The Solution Is: #x = {-1/2, -5/14}#

Isaiah Anthony · Answer

#x = -5/14, -1/2#

In order to solve this equation, first we will have to convert it into the general form, #ax^2 + bx + c = 0#, where #a != 0#.

So, #-16x^2 = 12x^2 + 24x + 5#

#rArr 28x^2 + 24x + 5 = 0# [Transposing #-16x^2# to the R.H.S]

We will now find the two roots using the Quadratic Formula, also known as Sridhar Acharya's Formula.

Here, Discriminant = #D# = #b^2 - 4ac = (24)^2 - 4 * 28 * 5#

#= 576 - 560 = 16 gt 0#

Thus, there will be two distinct and real roots to the equation.

Using the Formula Now,

#alpha# = #(- b + sqrt(D))/(2a) = (- 24 + sqrt(16))/(2 * 28) = -20/56 = -5/14#

and, #beta# = #(-b - sqrt(D))/(2a) = (- 24 - 4)/(2 * 28) = -28/56 = -1/2#

So, the two roots of the equation are #-5/14# and #-1/2#.

Eva Adamson · Answer

undefined To solve the equation -16x^2 = 12x^2 + 24x + 5, follow these steps:

1. Move all terms to one side of the equation to set it equal to zero:
   -16x^2 - 12x^2 - 24x - 5 = 0

2. Combine like terms:
   -28x^2 - 24x - 5 = 0

3. To solve this quadratic equation, you can use the quadratic formula:
   x = (-b ± √(b^2 - 4ac)) / (2a)

Substitute a = -28, b = -24, and c = -5 into the formula.

4. Calculate the discriminant (b^2 - 4ac):
   Discriminant = (-24)^2 - 4*(-28)*(-5) = 576 - 560 = 16

5. Since the discriminant is positive, there are two real solutions.

6. Plug the values into the quadratic formula:
   x = (-(-24) ± √16) / (2*(-28))
   x = (24 ± 4) / -56

7. Simplify:
   x1 = (24 + 4) / -56 = 28 / -56 = -1/2
   x2 = (24 - 4) / -56 = 20 / -56 = -5/14

Therefore, the solutions to the equation are x = -1/2 and x = -5/14.