2 Fragen zu Mathematics

To solve the simultaneous equations \(2x + y = 6\) and \(x^2 + y^2 = 72\), we can follow these steps: 1. **Express \(y\) in terms of \(x\)** from the first equation: \[ y = 6 - 2x \] 2. **Substitute \(y\) into the second equation**: \[ x^2 + (6 - 2x)^2 = 72 \] 3. **Expand the equation**: \[ x^2 + (36 - 24x + 4x^2) = 72 \] \[ 5x^2 - 24x + 36 = 72 \] 4. **Rearrange the equation**: \[ 5x^2 - 24x - 36 = 0 \] 5. **Use the quadratic formula** \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 5\), \(b = -24\), and \(c = -36\): \[ b^2 - 4ac = (-24)^2 - 4 \cdot 5 \cdot (-36) = 576 + 720 = 1296 \] \[ x = \frac{24 \pm \sqrt{1296}}{10} = \frac{24 \pm 36}{10} \] 6. **Calculate the two possible values for \(x\)**: - For \(x = \frac{60}{10} = 6\) - For \(x = \frac{-12}{10} = -1.2\) 7. **Find corresponding \(y\) values**: - If \(x = 6\): \[ y = 6 - 2(6) = 6 - 12 = -6 \] - If \(x = -1.2\): \[ y = 6 - 2(-1.2) = + 2.4 = 8.4 \] 8. **Solutions**: The solutions to the simultaneous equations are: \[ (x, y) = (6, -6) \quad \text{and} \quad (x, y) = (-1.2, 8.4) \]

"Rechenfaktor A für kubische Interpolation" wird auf Englisch als "Calculation factor A for cubic interpolation" übersetzt.