1. **Solve by graphing: $x^2 - 7x + 12 = 0$** - The equation is quadratic and can be written as $y = x^2 - 7x + 12$. - To solve by graphing, find the roots where $y=0$. - Factor: $x^2 - 7x + 12 = (x - 3)(x - 4)$. - Roots are $x=3$ and $x=4$. 2. **Solve by graphing: $x^2 + 12x = -36$** - Rewrite as $x^2 + 12x + 36 = 0$. - Recognize perfect square: $(x + 6)^2 = 0$. - Root is $x = -6$. 3. **Solve by graphing: $x + 1 = -x^2$** - Rewrite as $x^2 + x + 1 = 0$. - Calculate discriminant: $\Delta = 1^2 - 4(1)(1) = 1 - 4 = -3 < 0$. - No real roots; graph does not intersect x-axis. 4. **Solve using square roots: $14 = 2x^2$** - Divide both sides by 2: $$14 = 2x^2 \Rightarrow \cancel{2} \times 7 = \cancel{2} x^2 \Rightarrow 7 = x^2$$ - Take square root: $$x = \pm \sqrt{7}$$ 5. **Solve using square roots: $x^2 + 9 = 5$** - Subtract 9: $$x^2 = 5 - 9 = -4$$ - Since $x^2 = -4$ has no real solutions, no real roots. 6. **Solve using square roots: $(4x + 3)^2 = 16$** - Take square root: $$4x + 3 = \pm 4$$ - Case 1: $4x + 3 = 4 \Rightarrow 4x = 1 \Rightarrow x = \frac{1}{4}$ - Case 2: $4x + 3 = -4 \Rightarrow 4x = -7 \Rightarrow x = -\frac{7}{4}$ 7. **Solve by completing the square: $x^2 - 8x + 15 = 0$** - Move constant: $x^2 - 8x = -15$ - Complete square: add $(\frac{-8}{2})^2 = 16$ both sides $$x^2 - 8x + 16 = -15 + 16$$ $$(x - 4)^2 = 1$$ - Take square root: $$x - 4 = \pm 1$$ - Solutions: $x = 4 \pm 1 \Rightarrow x=5$ or $x=3$ 8. **Solve by completing the square: $x^2 - 6x = 10$** - Add $(\frac{-6}{2})^2 = 9$ both sides $$x^2 - 6x + 9 = 10 + 9$$ $$(x - 3)^2 = 19$$ - Take square root: $$x - 3 = \pm \sqrt{19}$$ - Solutions: $x = 3 \pm \sqrt{19}$ 9. **Solve by completing the square: $x^2 - 8x = -9$** - Add $(\frac{-8}{2})^2 = 16$ both sides $$x^2 - 8x + 16 = -9 + 16$$ $$(x - 4)^2 = 7$$ - Take square root: $$x - 4 = \pm \sqrt{7}$$ - Solutions: $x = 4 \pm \sqrt{7}$ 10. **Solve by completing the square: $16 = x^2 - 16x - 20$** - Move terms: $$x^2 - 16x - 20 - 16 = 0 \Rightarrow x^2 - 16x - 36 = 0$$ - Complete square: add $(\frac{-16}{2})^2 = 64$ both sides $$x^2 - 16x + 64 = 36 + 64$$ $$(x - 8)^2 = 100$$ - Take square root: $$x - 8 = \pm 10$$ - Solutions: $x = 8 \pm 10 \Rightarrow x=18$ or $x=-2$ 11. **Solve using quadratic formula: $5x^2 + x - 4 = 0$** - Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - Here $a=5$, $b=1$, $c=-4$ - Discriminant: $$\Delta = 1^2 - 4(5)(-4) = 1 + 80 = 81$$ - Roots: $$x = \frac{-1 \pm \sqrt{81}}{10} = \frac{-1 \pm 9}{10}$$ - Solutions: $$x = \frac{8}{10} = 0.8, \quad x = \frac{-10}{10} = -1$$ 12. **Solve using quadratic formula: $9x^2 + 6x + 1 = 0$** - $a=9$, $b=6$, $c=1$ - Discriminant: $$\Delta = 6^2 - 4(9)(1) = 36 - 36 = 0$$ - One root: $$x = \frac{-6}{2 \times 9} = \frac{-6}{18} = -\frac{1}{3}$$ 13. **Solve using quadratic formula: $-2x^2 + 3x + 7 = 0$** - $a=-2$, $b=3$, $c=7$ - Discriminant: $$\Delta = 3^2 - 4(-2)(7) = 9 + 56 = 65$$ - Roots: $$x = \frac{-3 \pm \sqrt{65}}{2(-2)} = \frac{-3 \pm \sqrt{65}}{-4}$$ 14. **Discriminant for $y = 4x^2 - 4x + 1$** - $a=4$, $b=-4$, $c=1$ - Discriminant: $$\Delta = (-4)^2 - 4(4)(1) = 16 - 16 = 0$$ - Graph touches x-axis once (one real root). 15. **Solve $x^2 - 9x - 10 = 0$ and choose method** - Try factoring: $$(x - 10)(x + 1) = 0$$ - Solutions: $x=10$ or $x=-1$ - Factoring is fastest here because coefficients are integers and easy to factor. 16. **Solve system:** $$y = x^2 - 4x - 2$$ $$y = -4x + 2$$ - Set equal: $$x^2 - 4x - 2 = -4x + 2$$ - Simplify: $$x^2 - 4x - 2 + 4x - 2 = 0 \Rightarrow x^2 - 4 = 0$$ - Solve: $$x^2 = 4 \Rightarrow x = \pm 2$$ - Find $y$: For $x=2$: $y = -4(2) + 2 = -8 + 2 = -6$ For $x=-2$: $y = -4(-2) + 2 = 8 + 2 = 10$ - Solutions: $(2, -6)$ and $(-2, 10)$ 17. **Solve system:** $$y = -5x^2 + x - 1$$ $$y = -7$$ - Set equal: $$-5x^2 + x - 1 = -7$$ - Simplify: $$-5x^2 + x - 1 + 7 = 0 \Rightarrow -5x^2 + x + 6 = 0$$ - Multiply by $-1$: $$5x^2 - x - 6 = 0$$ - Use quadratic formula: $$a=5, b=-1, c=-6$$ $$\Delta = (-1)^2 - 4(5)(-6) = 1 + 120 = 121$$ $$x = \frac{1 \pm 11}{10}$$ - Solutions: $$x = \frac{12}{10} = 1.2, \quad x = \frac{-10}{10} = -1$$ - Corresponding $y = -7$ - Solutions: $(1.2, -7)$ and $(-1, -7)$ 18. **Geometry: Area of triangle is 35 sq ft, base $= x + 4$, altitude $= x$** - Area formula: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ - Substitute: $$35 = \frac{1}{2} (x + 4)(x)$$ - Multiply both sides by 2: $$70 = x(x + 4) = x^2 + 4x$$ - Rearrange: $$x^2 + 4x - 70 = 0$$ - Use quadratic formula: $$a=1, b=4, c=-70$$ $$\Delta = 4^2 - 4(1)(-70) = 16 + 280 = 296$$ $$x = \frac{-4 \pm \sqrt{296}}{2}$$ - Approximate: $$\sqrt{296} \approx 17.2$$ $$x = \frac{-4 \pm 17.2}{2}$$ - Positive root: $$x = \frac{-4 + 17.2}{2} = \frac{13.2}{2} = 6.6$$ - Base length: $$x + 4 = 6.6 + 4 = 10.6$$ **Final answers:** 1. $x=3,4$ 2. $x=-6$ 3. No real solution 4. $x=\pm \sqrt{7}$ 5. No real solution 6. $x=\frac{1}{4}, -\frac{7}{4}$ 7. $x=3,5$ 8. $x=3 \pm \sqrt{19}$ 9. $x=4 \pm \sqrt{7}$ 10. $x=18, -2$ 11. $x=0.8, -1$ 12. $x=-\frac{1}{3}$ 13. $x=\frac{-3 \pm \sqrt{65}}{-4}$ 14. One intersection 15. $x=10, -1$ (factoring chosen) 16. $(2,-6), (-2,10)$ 17. $(1.2,-7), (-1,-7)$ 18. Base length $\approx 10.6$ ft