Simultaneous Equations 747222

1. **State the problem:** Solve the simultaneous equations: $$2y^2 + x^2 = -6x + 42$$ $$2x + y = -3$$ 2. **Express one variable in terms of the other:** From the second equation, $$2x + y = -3 \implies y = -3 - 2x$$ 3. **Substitute into the first equation:** Replace $y$ with $-3 - 2x$ in the first equation: $$2(-3 - 2x)^2 + x^2 = -6x + 42$$ 4. **Expand the square:** $$(-3 - 2x)^2 = (-3)^2 + 2 \times (-3) \times (-2x) + ( -2x)^2 = 9 + 12x + 4x^2$$ So, $$2(9 + 12x + 4x^2) + x^2 = -6x + 42$$ 5. **Distribute and simplify:** $$18 + 24x + 8x^2 + x^2 = -6x + 42$$ $$9x^2 + 24x + 18 = -6x + 42$$ 6. **Bring all terms to one side:** $$9x^2 + 24x + 18 + 6x - 42 = 0$$ $$9x^2 + 30x - 24 = 0$$ 7. **Simplify the equation by dividing by 3:** $$\cancel{3} \times 3x^2 + \cancel{3} \times 10x - \cancel{3} \times 8 = 0 \implies 3x^2 + 10x - 8 = 0$$ 8. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=10$, $c=-8$. Calculate the discriminant: $$\Delta = 10^2 - 4 \times 3 \times (-8) = 100 + 96 = 196$$ So, $$x = \frac{-10 \pm \sqrt{196}}{2 \times 3} = \frac{-10 \pm 14}{6}$$ 9. **Find the two values of $x$:** - For $+$: $$x = \frac{-10 + 14}{6} = \frac{4}{6} = \frac{2}{3}$$ - For $-$: $$x = \frac{-10 - 14}{6} = \frac{-24}{6} = -4$$ 10. **Find corresponding $y$ values:** Recall $y = -3 - 2x$. - For $x = \frac{2}{3}$: $$y = -3 - 2 \times \frac{2}{3} = -3 - \frac{4}{3} = -\frac{9}{3} - \frac{4}{3} = -\frac{13}{3}$$ - For $x = -4$: $$y = -3 - 2 \times (-4) = -3 + 8 = 5$$ 11. **Final solutions:** $$\boxed{\left(\frac{2}{3}, -\frac{13}{3}\right) \text{ and } (-4, 5)}$$