Question

factor each expression.

1. $x^2 - 4x - 45$

$(x - 9)(x + 5)$

1. $x^2 - 5x - 6$

$(x + 5)(x - 6)$

1. $x^2 - 17x + 72$

$(x - 9)(x - 8)$

1. $x^2 - 36$

$(x - 6)(x - 8)$

1. $x^2 + 7x - 8$

$(x - 1)(x + 8)$

1. $x^2 - 3x - 54$

$(x - $

1. $x^2 + 3x + 2$

$(x$

1. $x^2 + 3x - 18$

$($

1. $x^2 + 17x + 72$
2. $x^2 + 10x + 24$
3. $x^2 - 9x + 8$
4. $x^2 + x - 42$
5. $x^2 - x - 72$
6. $x^2 + 2x - 63$
7. $x^2 + 13x + 40$
8. $x^2 - 2x - 8$
9. $x^2 + x - 6$
10. $x^2 - 4x + 3$
11. $x^2 + 4x - 5$
12. $x^2 - 2x - 35$

Explanation:

Step1: Factor \(x^2 - 4x - 45\)

We need two numbers that multiply to \(-45\) and add to \(-4\). The numbers are \(-9\) and \(5\) since \(-9\times5 = -45\) and \(-9 + 5=-4\). So, \(x^2 - 4x - 45=(x - 9)(x + 5)\).

Step2: Factor \(x^2 - 5x - 6\)

Find two numbers that multiply to \(-6\) and add to \(-5\). The numbers are \(1\) and \(-6\) (wait, no, \(5\) and \(-6\): \(5\times(-6)=-30\)? Wait, no, original problem is \(x^2 - 5x - 6\). Wait, correct numbers: multiply to \(-6\), add to \(-5\). So \(-6\) and \(1\): \(-6\times1=-6\), \(-6 + 1=-5\). So \(x^2 - 5x - 6=(x + 1)(x - 6)\)? Wait, the handwritten answer is \((x + 5)(x - 6)\). Let's check: \((x + 5)(x - 6)=x^2 - 6x + 5x - 30=x^2 - x - 30\), which is wrong. Wait, maybe a typo in the problem? If the problem was \(x^2 + x - 30\), then \((x + 6)(x - 5)\)? Wait, no. Wait, the user's problem: "3. \(x^2 - 5x - 6\)" (wait, the numbering: first is 1, then 3? Maybe a typo. Anyway, following the handwritten, but let's do correctly. For \(x^2 - 5x - 6\), correct factors: find \(a,b\) with \(a\times b=-6\), \(a + b=-5\). So \(a=-6\), \(b = 1\). So \((x - 6)(x + 1)\). But the handwritten is \((x + 5)(x - 6)\), which is wrong. Maybe the problem was \(x^2 + x - 30\), then \((x + 6)(x - 5)\), no. Anyway, let's take problem 11: \(x^2 - 9x + 8\).

Step3: Factor \(x^2 - 9x + 8\)

Find two numbers that multiply to \(8\) and add to \(-9\). The numbers are \(-8\) and \(-1\) since \(-8\times(-1)=8\) and \(-8+(-1)=-9\). So \(x^2 - 9x + 8=(x - 8)(x - 1)\).

Step4: Factor \(x^2 - 36\)

This is a difference of squares: \(a^2 - b^2=(a - b)(a + b)\). Here, \(a = x\), \(b = 6\), so \(x^2 - 36=(x - 6)(x + 6)\). The handwritten is \((x - 6)(x - 8)\), which is wrong.

Step5: Factor \(x^2 + 7x - 8\)

Find two numbers that multiply to \(-8\) and add to \(7\). The numbers are \(8\) and \(-1\) since \(8\times(-1)=-8\) and \(8+(-1)=7\). So \(x^2 + 7x - 8=(x + 8)(x - 1)\). The handwritten is \((x - 1)(x + 8)\), which is correct (multiplication is commutative).

Step6: Factor \(x^2 - 3x - 54\)

Find two numbers that multiply to \(-54\) and add to \(-3\). The numbers are \(-9\) and \(6\) since \(-9\times6=-54\) and \(-9 + 6=-3\). So \(x^2 - 3x - 54=(x - 9)(x + 6)\).

Step7: Factor \(x^2 + 3x + 2\)

Find two numbers that multiply to \(2\) and add to \(3\). The numbers are \(1\) and \(2\) since \(1\times2 = 2\) and \(1+2 = 3\). So \(x^2 + 3x + 2=(x + 1)(x + 2)\).

Step8: Factor \(x^2 + 3x - 18\)

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

Step9: Factor \(x^2 + 17x + 72\)

Find two numbers that multiply to \(72\) and add to \(17\). The numbers are \(8\) and \(9\) since \(8\times9 = 72\) and \(8+9 = 17\). So \(x^2 + 17x + 72=(x + 8)(x + 9)\).

Step10: Factor \(x^2 + 10x + 24\)

Find two numbers that multiply to \(24\) and add to \(10\). The numbers are \(4\) and \(6\) since \(4\times6 = 24\) and \(4+6 = 10\). So \(x^2 + 10x + 24=(x + 4)(x + 6)\).

Step11: Factor \(x^2 - 9x + 8\) (problem 11)

As in step 3, \((x - 8)(x - 1)\).

Step12: Factor \(x^2 + x - 42\)

Find two numbers that multiply to \(-42\) and add to \(1\). The numbers are \(7\) and \(-6\) since \(7\times(-6)=-42\) and \(7+(-6)=1\). So \(x^2 + x - 42=(x + 7)(x - 6)\).

Step13: Factor \(x^2 - x - 72\)

Find two numbers that multiply to \(-72\) and add to \(-1\). The numbers are \(-9\) and \(8\) since \(-9\times8=-72\) and \(-9 + 8=-1\). So \(x^2 - x - 72=(x - 9)(x + 8)\).

Step14: Factor \(x^2 + 2x - 63\)

Find two numbers that multiply to \(-63\) and add to \(2\). The numbers are \(9\) and \(-7\) since \(9\times(-7)=-63\) and \(9+(-7)=2\). So \(x^2 + 2x - 63=(x + 9)(x - 7)\).

Step15: Factor \(x^2 + 13x + 40\)

Find two numbers that multiply to \(40\) and add to \(13\). The numbers are \(5\) and \(8\) since \(5\times8 = 40\) and \(5+8 = 13\). So \(x^2 + 13x + 40=(x + 5)(x + 8)\).

Step16: Factor \(x^2 - 2x - 8\)

Find two numbers that multiply to \(-8\) and add to \(-2\). The numbers are \(-4\) and \(2\) since \(-4\times2=-8\) and \(-4 + 2=-2\). So \(x^2 - 2x - 8=(x - 4)(x + 2)\).

Step17: Factor \(x^2 + x - 6\)

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

Step18: Factor \(x^2 - 4x + 3\)

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

Step19: Factor \(x^2 + 4x - 5\)

Find two numbers that multiply to \(-5\) and add to \(4\). The numbers are \(5\) and \(-1\) since \(5\times(-1)=-5\) and \(5+(-1)=4\). So \(x^2 + 4x - 5=(x + 5)(x - 1)\).

Step20: Factor \(x^2 - 2x - 35\)

Find two numbers that multiply to \(-35\) and add to \(-2\). The numbers are \(-7\) and \(5\) since \(-7\times5=-35\) and \(-7 + 5=-2\). So \(x^2 - 2x - 35=(x - 7)(x + 5)\).

Answer:

For each expression, the factored forms are:

1. \((x - 9)(x + 5)\)
2. (Correctly) \((x - 6)(x + 1)\) (but handwritten is wrong)
3. \((x - 9)(x - 8)\) (wait, problem 3: \(x^2 - 17x + 72\). Numbers: multiply to \(72\), add to \(-17\). So \(-8\) and \(-9\): \(-8\times(-9)=72\), \(-8+(-9)=-17\). So \(x^2 - 17x + 72=(x - 8)(x - 9)\), which matches the handwritten.
4. \((x - 6)(x + 6)\) (handwritten wrong)
5. \((x - 1)(x + 8)\)
6. \((x - 9)(x + 6)\)
7. \((x + 1)(x + 2)\)
8. \((x + 6)(x - 3)\)
9. \((x + 8)(x + 9)\)
10. \((x + 4)(x + 6)\)
11. \((x - 8)(x - 1)\)
12. \((x + 7)(x - 6)\)
13. \((x - 9)(x + 8)\)
14. \((x + 9)(x - 7)\)
15. \((x + 5)(x + 8)\)
16. \((x - 4)(x + 2)\)
17. \((x + 3)(x - 2)\)
18. \((x - 3)(x - 1)\)
19. \((x + 5)(x - 1)\)
20. \((x - 7)(x + 5)\)