Completing Square Ax2 B36B89

1. **Problem Statement:** Complete the square for each quadratic expression of the form $$ax^2 + bx + c$$ by factoring out the common factor from the first two terms, completing the square inside the bracket, and then multiplying back the common factor. 2. **Formula and Rules:** - For a quadratic $$ax^2 + bx + c$$, factor out $$a$$ from the first two terms: $$a(x^2 + \frac{b}{a}x) + c$$. - Complete the square inside the parentheses by adding and subtracting $$\left(\frac{b}{2a}\right)^2$$: $$a\left[\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right] + c$$. - Simplify the constants outside the square. 3. **Step-by-step solutions:** **(a) $$2x^2 + 16x$$** - Factor out 2: $$2[x^2 + 8x]$$ - Complete the square: $$2[(x + 4)^2 - 16]$$ because $$\left(\frac{8}{2}\right)^2 = 16$$ - Multiply out: $$2(x + 4)^2 - 32$$ **(b) $$3x^2 - 18x$$** - Factor out 3: $$3[x^2 - 6x]$$ - Complete the square: $$3[(x - 3)^2 - 9]$$ because $$\left(\frac{-6}{2}\right)^2 = 9$$ - Multiply out: $$3(x - 3)^2 - 27$$ **(c) $$2x^2 + 12x + 1$$** - Factor out 2: $$2[x^2 + 6x] + 1$$ - Complete the square: $$2[(x + 3)^2 - 9] + 1$$ because $$\left(\frac{6}{2}\right)^2 = 9$$ - Multiply out: $$2(x + 3)^2 - 18 + 1 = 2(x + 3)^2 - 17$$ **(d) $$2x^2 - 20x - 7$$** - Factor out 2: $$2[x^2 - 10x] - 7$$ - Complete the square: $$2[(x - 5)^2 - 25] - 7$$ because $$\left(\frac{-10}{2}\right)^2 = 25$$ - Multiply out: $$2(x - 5)^2 - 50 - 7 = 2(x - 5)^2 - 57$$ **(e) $$3x^2 + 6x - 5$$** - Factor out 3: $$3[x^2 + 2x] - 5$$ - Complete the square: $$3[(x + 1)^2 - 1] - 5$$ because $$\left(\frac{2}{2}\right)^2 = 1$$ - Multiply out: $$3(x + 1)^2 - 3 - 5 = 3(x + 1)^2 - 8$$ **(f) $$4x^2 + 16x - 1$$** - Factor out 4: $$4[x^2 + 4x] - 1$$ - Complete the square: $$4[(x + 2)^2 - 4] - 1$$ because $$\left(\frac{4}{2}\right)^2 = 4$$ - Multiply out: $$4(x + 2)^2 - 16 - 1 = 4(x + 2)^2 - 17$$ **(g) $$5x^2 - 30x + 11$$** - Factor out 5: $$5[x^2 - 6x] + 11$$ - Complete the square: $$5[(x - 3)^2 - 9] + 11$$ because $$\left(\frac{-6}{2}\right)^2 = 9$$ - Multiply out: $$5(x - 3)^2 - 45 + 11 = 5(x - 3)^2 - 34$$ **(h) $$2x^2 - 10x + 3$$ - Factor out 2: $$2[x^2 - 5x] + 3$$ - Complete the square: $$2\left[\left(x - \frac{5}{2}\right)^2 - \frac{25}{4}\right] + 3$$ because $$\left(\frac{-5}{2}\right)^2 = \frac{25}{4}$$ - Multiply out: $$2\left(x - \frac{5}{2}\right)^2 - 2 \times \frac{25}{4} + 3 = 2\left(x - \frac{5}{2}\right)^2 - \frac{25}{2} + 3$$ - Simplify constants: $$2\left(x - \frac{5}{2}\right)^2 - \frac{19}{2}$$ **(i) $$2x^2 + 6x - 1$$** - Factor out 2: $$2[x^2 + 3x] - 1$$ - Complete the square: $$2\left[(x + \frac{3}{2})^2 - \frac{9}{4}\right] - 1$$ because $$\left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ - Multiply out: $$2(x + \frac{3}{2})^2 - 2 \times \frac{9}{4} - 1 = 2(x + \frac{3}{2})^2 - \frac{9}{2} - 1$$ - Simplify constants: $$2(x + \frac{3}{2})^2 - \frac{11}{2}$$ **(j) $$3x^2 - 9x + 2$$ - Factor out 3: $$3[x^2 - 3x] + 2$$ - Complete the square: $$3\left[(x - \frac{3}{2})^2 - \frac{9}{4}\right] + 2$$ because $$\left(\frac{-3}{2}\right)^2 = \frac{9}{4}$$ - Multiply out: $$3(x - \frac{3}{2})^2 - 3 \times \frac{9}{4} + 2 = 3(x - \frac{3}{2})^2 - \frac{27}{4} + 2$$ - Simplify constants: $$3(x - \frac{3}{2})^2 - \frac{19}{4}$$ 4. **Summary:** Each quadratic is rewritten as $$a(x + h)^2 + k$$ where $$h = \frac{b}{2a}$$ and $$k = c - a\left(\frac{b}{2a}\right)^2$$. Final answers: - $$2(x + 4)^2 - 32$$ - $$3(x - 3)^2 - 27$$ - $$2(x + 3)^2 - 17$$ - $$2(x - 5)^2 - 57$$ - $$3(x + 1)^2 - 8$$ - $$4(x + 2)^2 - 17$$ - $$5(x - 3)^2 - 34$$ - $$2\left(x - \frac{5}{2}\right)^2 - \frac{19}{2}$$ - $$2\left(x + \frac{3}{2}\right)^2 - \frac{11}{2}$$ - $$3\left(x - \frac{3}{2}\right)^2 - \frac{19}{4}$$