1. **State the problem:** Solve the quadratic equation $$9x^2 + 7x - 56 = 0$$ using the x-method (factoring by grouping). 2. **Recall the formula and method:** For a quadratic equation $$ax^2 + bx + c = 0$$, the x-method involves finding two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate the product and sum:** Here, $$a = 9$$, $$b = 7$$, and $$c = -56$$. Calculate $$a \times c = 9 \times (-56) = -504$$. We need two numbers that multiply to $$-504$$ and add to $$7$$. 4. **Find the two numbers:** Factors of $$-504$$ that add to $$7$$ are $$21$$ and $$-24$$ because $$21 \times (-24) = -504$$ and $$21 + (-24) = -3$$ (not 7), so try again. Try $$-7$$ and $$72$$: $$-7 \times 72 = -504$$ and $$-7 + 72 = 65$$ (no). Try $$-9$$ and $$56$$: $$-9 \times 56 = -504$$ and $$-9 + 56 = 47$$ (no). Try $$-14$$ and $$36$$: $$-14 \times 36 = -504$$ and $$-14 + 36 = 22$$ (no). Try $$-18$$ and $$28$$: $$-18 \times 28 = -504$$ and $$-18 + 28 = 10$$ (no). Try $$-12$$ and $$42$$: $$-12 \times 42 = -504$$ and $$-12 + 42 = 30$$ (no). Try $$-21$$ and $$24$$: $$-21 \times 24 = -504$$ and $$-21 + 24 = 3$$ (no). Try $$-28$$ and $$18$$: $$-28 \times 18 = -504$$ and $$-28 + 18 = -10$$ (no). Try $$-36$$ and $$14$$: $$-36 \times 14 = -504$$ and $$-36 + 14 = -22$$ (no). Try $$-42$$ and $$12$$: $$-42 \times 12 = -504$$ and $$-42 + 12 = -30$$ (no). Try $$-56$$ and $$9$$: $$-56 \times 9 = -504$$ and $$-56 + 9 = -47$$ (no). Try $$-72$$ and $$7$$: $$-72 \times 7 = -504$$ and $$-72 + 7 = -65$$ (no). Try $$-84$$ and $$6$$: $$-84 \times 6 = -504$$ and $$-84 + 6 = -78$$ (no). Try $$-126$$ and $$4$$: $$-126 \times 4 = -504$$ and $$-126 + 4 = -122$$ (no). Try $$-168$$ and $$3$$: $$-168 \times 3 = -504$$ and $$-168 + 3 = -165$$ (no). Try $$-252$$ and $$2$$: $$-252 \times 2 = -504$$ and $$-252 + 2 = -250$$ (no). Try $$-504$$ and $$1$$: $$-504 \times 1 = -504$$ and $$-504 + 1 = -503$$ (no). Try $$24$$ and $$-21$$: $$24 \times (-21) = -504$$ and $$24 + (-21) = 3$$ (no). Try $$-24$$ and $$21$$: $$-24 \times 21 = -504$$ and $$-24 + 21 = -3$$ (no). Try $$-7$$ and $$72$$: $$-7 \times 72 = -504$$ and $$-7 + 72 = 65$$ (no). Try $$7$$ and $$-72$$: $$7 \times (-72) = -504$$ and $$7 + (-72) = -65$$ (no). Try $$-9$$ and $$56$$: $$-9 \times 56 = -504$$ and $$-9 + 56 = 47$$ (no). Try $$9$$ and $$-56$$: $$9 \times (-56) = -504$$ and $$9 + (-56) = -47$$ (no). Try $$-14$$ and $$36$$: $$-14 \times 36 = -504$$ and $$-14 + 36 = 22$$ (no). Try $$14$$ and $$-36$$: $$14 \times (-36) = -504$$ and $$14 + (-36) = -22$$ (no). Try $$-18$$ and $$28$$: $$-18 \times 28 = -504$$ and $$-18 + 28 = 10$$ (no). Try $$18$$ and $$-28$$: $$18 \times (-28) = -504$$ and $$18 + (-28) = -10$$ (no). Try $$-12$$ and $$42$$: $$-12 \times 42 = -504$$ and $$-12 + 42 = 30$$ (no). Try $$12$$ and $$-42$$: $$12 \times (-42) = -504$$ and $$12 + (-42) = -30$$ (no). Try $$-21$$ and $$24$$: $$-21 \times 24 = -504$$ and $$-21 + 24 = 3$$ (no). Try $$21$$ and $$-24$$: $$21 \times (-24) = -504$$ and $$21 + (-24) = -3$$ (no). Try $$-28$$ and $$18$$: $$-28 \times 18 = -504$$ and $$-28 + 18 = -10$$ (no). Try $$28$$ and $$-18$$: $$28 \times (-18) = -504$$ and $$28 + (-18) = 10$$ (no). Try $$-36$$ and $$14$$: $$-36 \times 14 = -504$$ and $$-36 + 14 = -22$$ (no). Try $$36$$ and $$-14$$: $$36 \times (-14) = -504$$ and $$36 + (-14) = 22$$ (no). Try $$-42$$ and $$12$$: $$-42 \times 12 = -504$$ and $$-42 + 12 = -30$$ (no). Try $$42$$ and $$-12$$: $$42 \times (-12) = -504$$ and $$42 + (-12) = 30$$ (no). Try $$-56$$ and $$9$$: $$-56 \times 9 = -504$$ and $$-56 + 9 = -47$$ (no). Try $$56$$ and $$-9$$: $$56 \times (-9) = -504$$ and $$56 + (-9) = 47$$ (no). Try $$-72$$ and $$7$$: $$-72 \times 7 = -504$$ and $$-72 + 7 = -65$$ (no). Try $$72$$ and $$-7$$: $$72 \times (-7) = -504$$ and $$72 + (-7) = 65$$ (no). 5. Since no integer pairs satisfy the conditions, the quadratic is not factorable by integers. 6. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substitute $$a=9$$, $$b=7$$, $$c=-56$$: $$x = \frac{-7 \pm \sqrt{7^2 - 4 \times 9 \times (-56)}}{2 \times 9} = \frac{-7 \pm \sqrt{49 + 2016}}{18} = \frac{-7 \pm \sqrt{2065}}{18}$$ 7. **Simplify the square root if possible:** $$2065 = 5 \times 413$$, no perfect square factors, so leave as is. 8. **Final answer:** $$x = \frac{-7 + \sqrt{2065}}{18}$$ or $$x = \frac{-7 - \sqrt{2065}}{18}$$ These are the two solutions to the quadratic equation.