Complete Square B5C905

1. **State the problem:** We need to solve the quadratic equation $$y = x^2 - 6x - 5$$ by completing the square. 2. **Recall the formula and rule:** To complete the square for an equation of the form $$x^2 + bx + c$$, we add and subtract $$\left(\frac{b}{2}\right)^2$$ inside the equation to form a perfect square trinomial. 3. **Rewrite the equation:** $$y = x^2 - 6x - 5$$ 4. **Complete the square:** Take half of the coefficient of $$x$$, which is $$-6$$, so half is $$\frac{-6}{2} = -3$$. Square it: $$(-3)^2 = 9$$. Add and subtract 9 inside the equation: $$y = (x^2 - 6x + 9) - 9 - 5$$ 5. **Factor the perfect square trinomial:** $$y = (x - 3)^2 - 14$$ 6. **Set $$y = 0$$ to find the roots:** $$0 = (x - 3)^2 - 14$$ 7. **Isolate the square term:** $$ (x - 3)^2 = 14 $$ 8. **Take the square root of both sides:** $$ x - 3 = \pm \sqrt{14} $$ 9. **Solve for $$x$$:** $$ x = 3 \pm \sqrt{14} $$ **Final answer:** $$ x = 3 \pm \sqrt{14} $$ This matches the form $$x = [?] \pm \sqrt{[ ]}$$ with $$3$$ in place of $$[?]$$ and $$14$$ in place of $$[ ]$$.