Question

lesson 1.3 practice b for use with the lesson \solve x² + bx + c = 0 by factoring\ factor the expression. if the expression cannot be factored, say so. 1. x² + 4x - 21 2. x² - 6x + 5 3. x² + 6x + 8 4. x² - x - 6 5. x² - x - 12 6. x² - 2x - 8 7. x² - 9x + 20 8. x² + 3x - 18 9. x² - 9 10. x² + 8x + 16 11. x² - 11x + 28 12. x² - 2x + 2 prime 13. x² + 4x - 32 14. x² - 3x - 10 15. x² - 25 16. x² - 9x + 14 17. x² - 100 18. x² - 8x - 15 prime solve the equation. 19. x² + x - 6 = 0 20. x² + 3x - 10 = 0 21. x² - 5x + 6 = 0 22. x² - 4x + 4 = 0 23. x² + 7x + 12 = 0 24. x² - 3x - 28 = 0 25. x² - 36 = 0 26. x² - 2x - 15 = 0 27. x² - 11x + 18 = 0 28. 3x² = 48 29. x² - 7x - 4 = -10 30. 9x - 8 = x² find the zeros of the function by rewriting the function in intercept form. 31. y = x² + 8x + 15 32. y = x² - 12x + 32 33. f(x) = x² - 2x - 35 34. y = x² - x - 30 35. g(x) = x² + 10x + 9 36. y = x² - 6x 37. h(x) = x² - 12x + 27 38. y = x² - 9 39. y = x² + 16x + 64

Explanation:

Step1: Factorize \(x^{2}+4x - 21\)

We need to find two numbers that multiply to \(-21\) and add up to \(4\). The numbers are \(7\) and \(- 3\). So \(x^{2}+4x - 21=(x + 7)(x-3)\)

Step2: Factorize \(x^{2}-6x + 5\)

Find two numbers that multiply to \(5\) and add up to \(-6\). The numbers are \(-5\) and \(-1\). So \(x^{2}-6x + 5=(x - 5)(x - 1)\)

Step3: Factorize \(x^{2}+6x + 8\)

Find two numbers that multiply to \(8\) and add up to \(6\). The numbers are \(2\) and \(4\). So \(x^{2}+6x + 8=(x + 2)(x + 4)\)

Step4: Factorize \(x^{2}-x - 6\)

Find two numbers that multiply to \(-6\) and add up to \(-1\). The numbers are \(-3\) and \(2\). So \(x^{2}-x - 6=(x-3)(x + 2)\)

Step5: Factorize \(x^{2}-x - 12\)

Find two numbers that multiply to \(-12\) and add up to \(-1\). The numbers are \(-4\) and \(3\). So \(x^{2}-x - 12=(x-4)(x + 3)\)

Step6: Factorize \(x^{2}-2x - 8\)

Find two numbers that multiply to \(-8\) and add up to \(-2\). The numbers are \(-4\) and \(2\). So \(x^{2}-2x - 8=(x-4)(x + 2)\)

Step7: Factorize \(x^{2}-9x + 20\)

Find two numbers that multiply to \(20\) and add up to \(-9\). The numbers are \(-5\) and \(-4\). So \(x^{2}-9x + 20=(x-5)(x - 4)\)

Step8: Factorize \(x^{2}+3x - 18\)

Find two numbers that multiply to \(-18\) and add up to \(3\). The numbers are \(6\) and \(-3\). So \(x^{2}+3x - 18=(x + 6)(x-3)\)

Step9: Factorize \(x^{2}-9\)

Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), here \(a=x\) and \(b = 3\), so \(x^{2}-9=(x + 3)(x-3)\)

Step10: Factorize \(x^{2}+8x + 16\)

Using the perfect - square formula \(a^{2}+2ab + b^{2}=(a + b)^{2}\), here \(a=x\) and \(b = 4\), so \(x^{2}+8x + 16=(x + 4)^{2}\)

Step11: Factorize \(x^{2}-11x + 28\)

Find two numbers that multiply to \(28\) and add up to \(-11\). The numbers are \(-7\) and \(-4\). So \(x^{2}-11x + 28=(x-7)(x - 4)\)

Step12: \(x^{2}-2x + 2\) is prime as there are no two real numbers that multiply to \(2\) and add up to \(-2\)

Step13: Factorize \(x^{2}+4x - 32\)

Find two numbers that multiply to \(-32\) and add up to \(4\). The numbers are \(8\) and \(-4\). So \(x^{2}+4x - 32=(x + 8)(x-4)\)

Step14: Factorize \(x^{2}-3x - 10\)

Find two numbers that multiply to \(-10\) and add up to \(-3\). The numbers are \(-5\) and \(2\). So \(x^{2}-3x - 10=(x-5)(x + 2)\)

Step15: Factorize \(x^{2}-25\)

Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), here \(a=x\) and \(b = 5\), so \(x^{2}-25=(x + 5)(x-5)\)

Step16: Factorize \(x^{2}-9x + 14\)

Find two numbers that multiply to \(14\) and add up to \(-9\). The numbers are \(-7\) and \(-2\). So \(x^{2}-9x + 14=(x-7)(x - 2)\)

Step17: Factorize \(x^{2}-100\)

Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), here \(a=x\) and \(b = 10\), so \(x^{2}-100=(x + 10)(x-10)\)

Step18: \(x^{2}-8x - 15\) is prime as there are no two real numbers that multiply to \(-15\) and add up to \(-8\)

For solving equations:

Step19: Solve \(x^{2}+x - 6=0\)

Factor the left - hand side: \((x + 3)(x-2)=0\). Then \(x+3 = 0\) or \(x - 2=0\), so \(x=-3\) or \(x = 2\)

Step20: Solve \(x^{2}+3x - 10=0\)

Factor the left - hand side: \((x + 5)(x-2)=0\). Then \(x+5 = 0\) or \(x - 2=0\), so \(x=-5\) or \(x = 2\)

Step21: Solve \(x^{2}-5x + 6=0\)

Factor the left - hand side: \((x-2)(x - 3)=0\). Then \(x-2 = 0\) or \(x - 3=0\), so \(x=2\) or \(x = 3\)

Step22: Solve \(x^{2}-4x + 4=0\)

Factor the left - hand side: \((x - 2)^{2}=0\), so \(x = 2\)

Step23: Solve \(x^{2}+7x + 12=0\)

Factor the left - hand side: \((x + 3)(x + 4)=0\). Then \(x+3 = 0\) or \(x + 4=0\), so \(x=-3\) or \(x=-4\)

Step24: Solve \(x^{2}-3x - 28=0\)

Factor the left - hand side: \((x-7)(x + 4)=0\). Then \(x-7 = 0\) or \(x + 4=0\), so \(x=7\) or \(x=-4\)

Step25: Solve \(x^{2}-36=0\)

Factor the left - hand side: \((x + 6)(x-6)=0\). Then \(x+6 = 0\) or \(x - 6=0\), so \(x=-6\) or \(x = 6\)

Step26: Solve \(x^{2}-2x - 15=0\)

Factor the left - hand side: \((x-5)(x + 3)=0\). Then \(x-5 = 0\) or \(x + 3=0\), so \(x=5\) or \(x=-3\)

Step27: Solve \(x^{2}-11x + 18=0\)

Factor the left - hand side: \((x-2)(x - 9)=0\). Then \(x-2 = 0\) or \(x - 9=0\), so \(x=2\) or \(x = 9\)

Step28: Solve \(3x^{2}=48\)

First, rewrite it as \(x^{2}=16\), then \(x^{2}-16=0\). Factor the left - hand side: \((x + 4)(x-4)=0\), so \(x=-4\) or \(x = 4\)

Step29: Solve \(x^{2}-7x-4=-10\)

Rewrite it as \(x^{2}-7x + 6=0\). Factor the left - hand side: \((x-1)(x - 6)=0\). Then \(x-1 = 0\) or \(x - 6=0\), so \(x=1\) or \(x = 6\)

Step30: Solve \(9x-8=x^{2}\)

Rewrite it as \(x^{2}-9x + 8=0\). Factor the left - hand side: \((x-1)(x - 8)=0\). Then \(x-1 = 0\) or \(x - 8=0\), so \(x=1\) or \(x = 8\)

For finding zeros:

Step31: For \(y=x^{2}+8x + 15\)

Factor: \(y=(x + 3)(x + 5)\). Set \(y = 0\), then \(x=-3\) or \(x=-5\)

Step32: For \(y=x^{2}-12x + 32\)

Factor: \(y=(x-4)(x - 8)\). Set \(y = 0\), then \(x=4\) or \(x = 8\)

Step33: For \(f(x)=x^{2}-2x - 35\)

Factor: \(f(x)=(x-7)(x + 5)\). Set \(f(x)=0\), then \(x=7\) or \(x=-5\)

Step34: For \(y=x^{2}-x - 30\)

Factor: \(y=(x-6)(x + 5)\). Set \(y = 0\), then \(x=6\) or \(x=-5\)

Step35: For \(g(x)=x^{2}+10x + 9\)

Factor: \(g(x)=(x + 1)(x + 9)\). Set \(g(x)=0\), then \(x=-1\) or \(x=-9\)

Step36: For \(y=x^{2}-6x\)

Factor: \(y=x(x - 6)\). Set \(y = 0\), then \(x=0\) or \(x = 6\)

Step37: For \(h(x)=x^{2}-12x + 27\)

Factor: \(h(x)=(x-3)(x - 9)\). Set \(h(x)=0\), then \(x=3\) or \(x = 9\)

Step38: For \(y=x^{2}-9\)

Factor: \(y=(x + 3)(x-3)\). Set \(y = 0\), then \(x=-3\) or \(x = 3\)

Step39: For \(y=x^{2}+16x + 64\)

Factor: \(y=(x + 8)^{2}\). Set \(y = 0\), then \(x=-8\)

Answer:

1. \((x + 7)(x-3)\)
2. \((x - 5)(x - 1)\)
3. \((x + 2)(x + 4)\)
4. \((x-3)(x + 2)\)
5. \((x-4)(x + 3)\)
6. \((x-4)(x + 2)\)
7. \((x-5)(x - 4)\)
8. \((x + 6)(x-3)\)
9. \((x + 3)(x-3)\)
10. \((x + 4)^{2}\)
11. \((x-7)(x - 4)\)
12. Prime
13. \((x + 8)(x-4)\)
14. \((x-5)(x + 2)\)
15. \((x + 5)(x-5)\)
16. \((x-7)(x - 2)\)
17. \((x + 10)(x-10)\)
18. Prime
19. \(x=-3,x = 2\)
20. \(x=-5,x = 2\)
21. \(x=2,x = 3\)
22. \(x = 2\)
23. \(x=-3,x=-4\)
24. \(x=7,x=-4\)
25. \(x=-6,x = 6\)
26. \(x=5,x=-3\)
27. \(x=2,x = 9\)
28. \(x=-4,x = 4\)
29. \(x=1,x = 6\)
30. \(x=1,x = 8\)
31. \(x=-3,x=-5\)
32. \(x=4,x = 8\)
33. \(x=7,x=-5\)
34. \(x=6,x=-5\)
35. \(x=-1,x=-9\)
36. \(x=0,x = 6\)
37. \(x=3,x = 9\)
38. \(x=-3,x = 3\)
39. \(x=-8\)