5. $$ 4 x^{2}+24 x+16=50 .\left(\frac{6}{2}\right)^{2}=(3)^{2}=9 $$

Question

UpStudy AI · Accepted Answer

We start with the equation

$$
4x^2 + 24x + 16 = 50.
$$

**Step 1. Divide by 4.**

It is easier to complete the square when the coefficient of $$ x^2 $$ is 1. Divide every term by 4:

$$
x^2 + 6x + 4 = \frac{50}{4} = \frac{25}{2}.
$$

**Step 2. Isolate the quadratic and linear terms.**

Subtract 4 from both sides:

$$
x^2 + 6x = \frac{25}{2} - 4 = \frac{25}{2} - \frac{8}{2} = \frac{17}{2}.
$$

**Step 3. Complete the square.**

To complete the square for $$ x^2 + 6x $$, we need to add

$$
\left(\frac{6}{2}\right)^2 = 3^2 = 9.
$$

Add 9 to both sides:

$$
x^2 + 6x + 9 = \frac{17}{2} + 9.
$$

Now the left side is a perfect square:

$$
(x+3)^2 = \frac{17}{2} + \frac{18}{2} = \frac{35}{2}.
$$

**Step 4. Solve for $$ x $$.**

Take the square root of both sides:

$$
x+3 = \pm \sqrt{\frac{35}{2}}.
$$

Thus,

$$
x = -3 \pm \sqrt{\frac{35}{2}}.
$$

For a more simplified radical form, rationalize the denominator:

$$
\sqrt{\frac{35}{2}} = \frac{\sqrt{70}}{2},
$$

so the solutions can also be written as

$$
x = -3 \pm \frac{\sqrt{70}}{2}.
$$
The solutions are $$ x = -3 + \frac{\sqrt{70}}{2} $$ and $$ x = -3 - \frac{\sqrt{70}}{2} $$.

5. \( 4 x^{2}+24 x+16=50 .\left(\frac{6}{2}\right)^{2}=(3)^{2}=9 \)