find the value of x.\nx = □

Explanation:

Step1: Apply the secant - secant rule

If two secants are drawn to a circle from an exterior point, then $(a)(a + b)=(c)(c + d)$. Here, $a = 3$, $b = 5$, $c=x$, $d = 2$. So, $3\times(3 + 5)=x\times(x + 2)$.

Step2: Simplify the left - hand side

$3\times(3 + 5)=3\times8 = 24$. So the equation becomes $24=x(x + 2)=x^{2}+2x$.

Step3: Rearrange to quadratic form

$x^{2}+2x-24 = 0$.

Step4: Factor the quadratic equation

We need to find two numbers that multiply to - 24 and add up to 2. The numbers are 6 and - 4. So, $x^{2}+2x - 24=(x + 6)(x - 4)=0$.

Step5: Solve for x

Setting each factor equal to zero gives $x+6 = 0$ or $x - 4=0$. So $x=-6$ or $x = 4$. But since $x$ represents a length, we discard the negative value.

Answer:

$4$