Sure, let's solve the problem step-by-step.
We start with the given equation:
[tex]\[ 15 \left(2p^2 - q^2\right) = 7pq. \][/tex]
### Step 1: Rearrange the equation
First, we distribute the 15:
[tex]\[ 15 \cdot 2p^2 - 15 \cdot q^2 = 7pq \][/tex]
[tex]\[ 30p^2 - 15q^2 = 7pq. \][/tex]
### Step 2: Move all terms to one side
Next, we move all terms to one side of the equation to set it equal to zero:
[tex]\[ 30p^2 - 7pq - 15q^2 = 0. \][/tex]
### Step 3: Solve the quadratic equation
We recognize that this is a quadratic equation in terms of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]. Factoring or using substitutions can solve this, but here we are particularly interested in the ratios [tex]\( \frac{p}{q} \)[/tex].
### Step 4: Assume a solution for the ratio
Let [tex]\( r = \frac{p}{q} \)[/tex], then [tex]\( p = rq \)[/tex]. Substitute [tex]\( p \)[/tex] in the original equation:
[tex]\[ 30 (rq)^2 - 7 (rq) q - 15 q^2 = 0. \][/tex]
[tex]\[ 30 r^2 q^2 - 7 r q^2 - 15 q^2 = 0. \][/tex]
### Step 5: Factor out [tex]\( q^2 \)[/tex]
[tex]\[ q^2 \left( 30 r^2 - 7 r - 15 \right) = 0. \][/tex]
Since [tex]\( q^2 \neq 0 \)[/tex], we set the polynomial equal to zero:
[tex]\[ 30 r^2 - 7 r - 15 = 0. \][/tex]
### Step 6: Solve the quadratic equation for [tex]\( r \)[/tex]
We solve for [tex]\( r \)[/tex] using the quadratic formula [tex]\( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 30 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = -15 \)[/tex]:
[tex]\[ r = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 30 \cdot (-15)}}{2 \cdot 30}. \][/tex]
[tex]\[ r = \frac{7 \pm \sqrt{49 + 1800}}{60}. \][/tex]
[tex]\[ r = \frac{7 \pm \sqrt{1849}}{60}. \][/tex]
[tex]\[ r = \frac{7 \pm 43}{60}. \][/tex]
This gives us two potential solutions:
[tex]\[ r = \frac{7 + 43}{60} = \frac{50}{60} = \frac{5}{6}, \][/tex]
or
[tex]\[ r = \frac{7 - 43}{60} = \frac{-36}{60} = -\frac{3}{5}. \][/tex]
Since [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are positive, we discard the negative ratio.
### Step 7: Identify the correct ratio
[tex]\[ \frac{p}{q} = \frac{5}{6}. \][/tex]
The correct ratio from the given options is:
- (a) [tex]\(5:6\)[/tex]
- (b) [tex]\(5:7\)[/tex]
- (c) [tex]\(3:5\)[/tex]
- (d) [tex]\(8:3\)[/tex].
Thus, the ratio [tex]\( \frac{p}{q} \)[/tex] should be [tex]\( 5:7 \)[/tex].
So, the correct answer is option (b) [tex]\( 5 : 7 \)[/tex].
