Question

solve the following quadratic equation for all values of $x$ in simplest form.
$25(5x - 3)^2 + 6 = 22$
answer attempt 1 out of 2
additional solution no solution
$x =$

Explanation:

Step1: Isolate the squared term

Subtract 6 from both sides:
$25(5x-3)^2 = 22 - 6$
$25(5x-3)^2 = 16$

Step2: Divide by 25

Divide both sides by 25:
$(5x-3)^2 = \frac{16}{25}$

Step3: Take square root of both sides

Take square root of both sides:
$5x-3 = \pm\sqrt{\frac{16}{25}}$
$5x-3 = \pm\frac{4}{5}$

Step4: Solve for x (positive case)

Add 3 to both sides, then divide by 5:
$5x = 3 + \frac{4}{5}$
$5x = \frac{15}{5} + \frac{4}{5} = \frac{19}{5}$
$x = \frac{19}{25}$

Step5: Solve for x (negative case)

Add 3 to both sides, then divide by 5:
$5x = 3 - \frac{4}{5}$
$5x = \frac{15}{5} - \frac{4}{5} = \frac{11}{5}$
$x = \frac{11}{25}$

Answer:

$x=\frac{17}{25}$ and $x=\frac{13}{25}$