Quadratic Equation 2B5861

1. **State the problem:** Find the quadratic equation $y = ax^2 + bx + c$ passing through points $(-2,43)$, $(1,10)$, and $(5,22)$. 2. **Use the general quadratic form:** $y = ax^2 + bx + c$. We need to find $a$, $b$, and $c$. 3. **Set up equations using each point:** - For $(-2,43)$: $43 = a(-2)^2 + b(-2) + c = 4a - 2b + c$ - For $(1,10)$: $10 = a(1)^2 + b(1) + c = a + b + c$ - For $(5,22)$: $22 = a(5)^2 + b(5) + c = 25a + 5b + c$ 4. **Write the system of equations:** $$\begin{cases} 4a - 2b + c = 43 \\ a + b + c = 10 \\ 25a + 5b + c = 22 \end{cases}$$ 5. **Subtract the second equation from the first and third to eliminate $c$: - $(4a - 2b + c) - (a + b + c) = 43 - 10$ - $(25a + 5b + c) - (a + b + c) = 22 - 10$ This gives: - $4a - 2b + c - a - b - c = 33 \Rightarrow 3a - 3b = 33$ - $25a + 5b + c - a - b - c = 12 \Rightarrow 24a + 4b = 12$ 6. **Simplify these two equations:** - $3a - 3b = 33$ - $24a + 4b = 12$ 7. **Divide the first by 3 and the second by 4:** - $a - b = 11$ - $6a + b = 3$ 8. **Add the two equations to eliminate $b$: $$ (a - b) + (6a + b) = 11 + 3 \Rightarrow 7a = 14 \Rightarrow a = 2 $$ 9. **Substitute $a=2$ into $a - b = 11$: $$ 2 - b = 11 \Rightarrow b = 2 - 11 = -9 $$ 10. **Substitute $a=2$ and $b=-9$ into $a + b + c = 10$: $$ 2 + (-9) + c = 10 \Rightarrow -7 + c = 10 \Rightarrow c = 17 $$ 11. **Write the quadratic equation:** $$ y = 2x^2 - 9x + 17 $$ **Final answer:** The quadratic equation passing through the points is $$y = 2x^2 - 9x + 17$$.