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

2 Answers
Jan 15, 2018

x = -5/14, -1/2

Explanation:

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]

Now, we will use the Quadratic Formula or Sridhar Acharya's Formula, to solve for the two roots.

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

= 576 - 560 = 16 gt 0

So, the equation will have two real and distinct roots.

Now, Applying the Formula,

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.

See a solution process below:

Explanation:

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}