Answer :
To solve the equation [tex]\( p = q^2 - p^2 \)[/tex] given that [tex]\( p = 3 \)[/tex] and [tex]\( q = 1 \)[/tex], we can follow a step-by-step approach:
1. Substitute the Value of [tex]\( q \)[/tex]:
- From the question, we have [tex]\( q = 1 \)[/tex]. Substitute this into the equation:
[tex]\[
p = 1^2 - p^2
\][/tex]
This simplifies to:
[tex]\[
p = 1 - p^2
\][/tex]
2. Rearrange the Equation:
- Bring all terms to one side of the equation:
[tex]\[
p^2 + p - 1 = 0
\][/tex]
3. Solve the Quadratic Equation:
- This is a quadratic equation in standard form, [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -1 \)[/tex].
4. Use the Quadratic Formula:
- The quadratic formula is given by:
[tex]\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Calculate the Discriminant:
- The discriminant [tex]\( D \)[/tex] is:
[tex]\[
D = b^2 - 4ac = 1^2 - 4 \times 1 \times (-1) = 1 + 4 = 5
\][/tex]
6. Find the Roots:
- Substitute the discriminant back into the quadratic formula:
[tex]\[
p = \frac{-1 \pm \sqrt{5}}{2}
\][/tex]
- This gives two potential solutions for [tex]\( p \)[/tex]:
- First root:
[tex]\[
p = \frac{-1 - \sqrt{5}}{2} \approx -1.618033988749895
\][/tex]
- Second root:
[tex]\[
p = \frac{-1 + \sqrt{5}}{2} \approx 0.6180339887498949
\][/tex]
These two values are the solutions to the equation under the given conditions.
1. Substitute the Value of [tex]\( q \)[/tex]:
- From the question, we have [tex]\( q = 1 \)[/tex]. Substitute this into the equation:
[tex]\[
p = 1^2 - p^2
\][/tex]
This simplifies to:
[tex]\[
p = 1 - p^2
\][/tex]
2. Rearrange the Equation:
- Bring all terms to one side of the equation:
[tex]\[
p^2 + p - 1 = 0
\][/tex]
3. Solve the Quadratic Equation:
- This is a quadratic equation in standard form, [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -1 \)[/tex].
4. Use the Quadratic Formula:
- The quadratic formula is given by:
[tex]\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Calculate the Discriminant:
- The discriminant [tex]\( D \)[/tex] is:
[tex]\[
D = b^2 - 4ac = 1^2 - 4 \times 1 \times (-1) = 1 + 4 = 5
\][/tex]
6. Find the Roots:
- Substitute the discriminant back into the quadratic formula:
[tex]\[
p = \frac{-1 \pm \sqrt{5}}{2}
\][/tex]
- This gives two potential solutions for [tex]\( p \)[/tex]:
- First root:
[tex]\[
p = \frac{-1 - \sqrt{5}}{2} \approx -1.618033988749895
\][/tex]
- Second root:
[tex]\[
p = \frac{-1 + \sqrt{5}}{2} \approx 0.6180339887498949
\][/tex]
These two values are the solutions to the equation under the given conditions.
