Question

How do you solve using the completing the square method `3x^2+10x+13`?

Answer 1

Required Solutions are

`color(blue)(x = (-5/3) +- [sqrt(14)/3]i`

Explanation:

We are given the quadratic expression

`color(red)(f(x)=3x^2+10x+13`

We must use Completing The Square Method to find the solutions.

Let us assume that `color(red)(" " 3x^2+10x+13 = 0)`

We will find the solutions in steps.

`color(green)(Step.1)`

In this step,

we will move the constant term to Right-Hand Side (RHS)

Hence, we get

`color(red)(" " 3x^2+10x = -13)`

`color(green)(Step.2)`

Divide each term by 3 to get the coefficient of the `x^2` term as `1`

`(3/3)x^2+(10/3)x=(-13/3)`

`rArr x^2 + (10/3)x = (-13/3)`

`color(green)(Step.3)`

We will add a value to each side

`rArr x^2 + (10/3)x+ color(red)square = -(13/3) + color(red)square`

In the next step, we need to figure out how-to find the value that goes into the `color(red)(RED)` box

`color(green)(Step.4)`

Divide the coefficient of the `x` term by `2` and square it.

The result of the calculation will replace the `color(red)(RED)` box in the next step.

The calculation is done as follows:

Coefficient of the x-term is `(10/3)`

If we divide by 2, we will get `(10/6)`

When we square this intermediate result, we get

`(10/6)^2 rArr (5/3)^2 rArr (25/9)`

Hence, the value `(25/9)` will replace the `color(red)(RED)` box in the next step.

`color(green)(Step.5)`

`rArr x^2 + (10/3)x+ (25/9) = -(13/3) + (25/9)`

We can write the Left-Hand-Side (LHS) as

`rArr (x+color(red)(5/3))^2 = (-39 + 25)/9`

Note that we get the value `color(red)(5/3)` by dividing the coefficient of the x-term by `2`

`color(green)(Step.6)`

We will now work on

`(x+color(red)(5/3))^2 = (-39 + 25)/9` **

We will take Square Root on both sides to simplify:

Hence, we get

`sqrt[x+(5/3)^2] = +-sqrt[(-39 + 25)/9]`

Square root and Square cancel out.

`:.` on simplification we get,

`x+5/3 = +-sqrt[(-14)/9]`

`rArr x+5/3 = +-sqrt(14*(-1))/sqrt(9)`

Note that, in complex number system,

`i = sqrt(-1)`

`i^2 = i*i= (-1)`

Hence,

`rArr x+5/3 = +-sqrt(14*i^2)/3`

`x = -5/3 +-[sqrt(14)/3]i`

Hence, required solutions are

`color(blue)(x = (-5/3) +- [sqrt(14)/3]i`

Hope this helps.