Question

factor each completely.

1. $x^{2}-2x - 48$
2. $n^{2}+8n - 20$
3. $x^{2}+15x + 56$
4. $x^{2}-12x + 27$
5. $r^{2}-r - 20$
6. $n^{2}+7n - 30$
7. $k^{2}+15k + 50$
8. $x^{2}+16x + 60$

Explanation:

Step1: For quadratic $ax^{2}+bx + c$ find two numbers

For each quadratic, we need to find two numbers that multiply to $ac$ and add up to $b$.

Step2: Factor for 9) $x^{2}-2x - 48$

We need two numbers that multiply to $-48$ and add to $-2$. The numbers are $-8$ and $6$. So $x^{2}-2x - 48=x^{2}-8x+6x - 48=x(x - 8)+6(x - 8)=(x - 8)(x + 6)$.

Step3: Factor for 10) $n^{2}+8n - 20$

Two numbers that multiply to $-20$ and add to $8$ are $10$ and $-2$. So $n^{2}+8n - 20=n^{2}+10n-2n - 20=n(n + 10)-2(n + 10)=(n + 10)(n - 2)$.

Step4: Factor for 11) $x^{2}+15x + 56$

Numbers that multiply to $56$ and add to $15$ are $7$ and $8$. So $x^{2}+15x + 56=x^{2}+7x+8x + 56=x(x + 7)+8(x + 7)=(x + 7)(x + 8)$.

Step5: Factor for 12) $x^{2}-12x + 27$

Numbers that multiply to $27$ and add to $-12$ are $-3$ and $-9$. So $x^{2}-12x + 27=x^{2}-3x-9x + 27=x(x - 3)-9(x - 3)=(x - 3)(x - 9)$.

Step6: Factor for 13) $r^{2}-r - 20$

Numbers that multiply to $-20$ and add to $-1$ are $-5$ and $4$. So $r^{2}-r - 20=r^{2}-5r+4r - 20=r(r - 5)+4(r - 5)=(r - 5)(r + 4)$.

Step7: Analyze 14) $n^{2}+7n - 30$

We look for two integers that multiply to $-30$ and add to $7$. There are no such integers, so it cannot be factored over the integers.

Step8: Factor for 15) $k^{2}+15k + 50$

Numbers that multiply to $50$ and add to $15$ are $5$ and $10$. So $k^{2}+15k + 50=k^{2}+5k+10k + 50=k(k + 5)+10(k + 5)=(k + 5)(k + 10)$.

Step9: Factor for 16) $x^{2}+16x + 60$

Numbers that multiply to $60$ and add to $16$ are $6$ and $10$. So $x^{2}+16x + 60=x^{2}+6x+10x + 60=x(x + 6)+10(x + 6)=(x + 6)(x + 10)$.

Answer:

1. $(x - 8)(x+6)$
2. $(n + 10)(n - 2)$
3. $(x + 7)(x+8)$
4. $(x - 3)(x - 9)$
5. $(r - 5)(r + 4)$
6. This cannot be factored over the integers
7. $(k + 5)(k+10)$
8. $(x + 6)(x + 10)$