Sure, let's break this down step-by-step:
### Using Pythagoras' Theorem
1. Given Data:
- Side [tex]\( a = 9 \)[/tex]
- Side [tex]\( b = 5 \)[/tex]
2. Finding Side [tex]\( c \)[/tex]:
- According to the Pythagorean theorem: [tex]\( a^2 = b^2 + c^2 \)[/tex]
- Substituting the known values:
[tex]\[
9^2 = 5^2 + c^2
\][/tex]
[tex]\[
81 = 25 + c^2
\][/tex]
- Solving for [tex]\( c^2 \)[/tex]:
[tex]\[
c^2 = 81 - 25
\][/tex]
[tex]\[
c^2 = 56
\][/tex]
- Taking the square root of both sides to find [tex]\( c \)[/tex]:
[tex]\[
c = \sqrt{56}
\][/tex]
Simplifying [tex]\( \sqrt{56} \)[/tex]:
[tex]\[
c \approx 7.483314773547883
\][/tex]
### Solving the Quadratic Equation [tex]\( 2x^2 - 4x - 9 = 0 \)[/tex]
1. Formulating the Equation:
- The quadratic equation given is: [tex]\( 2x^2 - 4x - 9 = 0 \)[/tex]
2. Solving the Quadratic Equation:
- We use the quadratic formula: [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]
- Here, [tex]\( a = 2 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -9 \)[/tex]
- Substituting these values into the quadratic formula:
[tex]\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-9)}}{2 \cdot 2}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{16 + 72}}{4}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{88}}{4}
\][/tex]
[tex]\[
x = \frac{4 \pm 2\sqrt{22}}{4}
\][/tex]
[tex]\[
x = 1 \pm \frac{\sqrt{22}}{2}
\][/tex]
- Therefore, the solutions to the quadratic equation [tex]\( 2x^2 - 4x - 9 = 0 \)[/tex] are:
[tex]\[
x = 1 - \frac{\sqrt{22}}{2} \quad \text{and} \quad x = 1 + \frac{\sqrt{22}}{2}
\][/tex]
### Final Answer:
- The length of the side [tex]\( c \)[/tex] in the triangle calculated using the Pythagorean theorem is approximately [tex]\( 7.483314773547883 \)[/tex].
- The solutions to the quadratic equation [tex]\( 2x^2 - 4x - 9 = 0 \)[/tex] are [tex]\( x = 1 - \frac{\sqrt{22}}{2} \)[/tex] and [tex]\( x = 1 + \frac{\sqrt{22}}{2} \)[/tex].
