Question

solve each equation by factoring.

1. $3m^2 + 5m - 8 = 0$
2. $3k^2 + 7k - 6 = 0$

1. $3x^2 + 22x + 7 = 0$
2. $7r^2 + 10r + 3 = 0$

1. $5v^2 + 16v - 16 = 0$
2. $21a^2 + 41a = 40$

1. $7v^2 + 19v = 6$
2. $7b^2 + 21 = 52b$

1. $3a^2 = 56 - 17a$
2. $2n^2 = -n$

Explanation:

1) Solve \(3m^{2}+5m - 8 = 0\)

Step 1: Factor the quadratic

We need two numbers that multiply to \(3\times(-8)=-24\) and add up to \(5\). The numbers are \(8\) and \(-3\).
Rewrite the middle term: \(3m^{2}+8m - 3m - 8 = 0\)
Group the terms: \((3m^{2}+8m)+(-3m - 8)=0\)
Factor out the GCF from each group: \(m(3m + 8)-1(3m + 8)=0\)
Factor out \((3m + 8)\): \((3m + 8)(m - 1)=0\)

Step 2: Solve for \(m\)

Set each factor equal to zero:
\(3m+8 = 0\) or \(m - 1 = 0\)
For \(3m+8 = 0\): \(3m=-8\Rightarrow m=-\frac{8}{3}\)
For \(m - 1 = 0\): \(m = 1\)

2) Solve \(3k^{2}+7k - 6 = 0\)

Step 1: Factor the quadratic

Find two numbers that multiply to \(3\times(-6)=-18\) and add up to \(7\). The numbers are \(9\) and \(-2\).
Rewrite the middle term: \(3k^{2}+9k - 2k - 6 = 0\)
Group the terms: \((3k^{2}+9k)+(-2k - 6)=0\)
Factor out the GCF: \(3k(k + 3)-2(k + 3)=0\)
Factor out \((k + 3)\): \((3k - 2)(k + 3)=0\)

Step 2: Solve for \(k\)

Set each factor equal to zero:
\(3k - 2 = 0\) or \(k + 3 = 0\)
For \(3k - 2 = 0\): \(3k=2\Rightarrow k=\frac{2}{3}\)
For \(k + 3 = 0\): \(k=-3\)

3) Solve \(3x^{2}+22x + 7 = 0\)

Step 1: Factor the quadratic

We need two numbers that multiply to \(3\times7 = 21\) and add up to \(22\). The numbers are \(21\) and \(1\).
Rewrite the middle term: \(3x^{2}+21x+x + 7 = 0\)
Group the terms: \((3x^{2}+21x)+(x + 7)=0\)
Factor out the GCF: \(3x(x + 7)+1(x + 7)=0\)
Factor out \((x + 7)\): \((3x + 1)(x + 7)=0\)

Step 2: Solve for \(x\)

Set each factor equal to zero:
\(3x+1 = 0\) or \(x + 7 = 0\)
For \(3x+1 = 0\): \(3x=-1\Rightarrow x=-\frac{1}{3}\)
For \(x + 7 = 0\): \(x=-7\)

4) Solve \(7r^{2}+10r + 3 = 0\)

Step 1: Factor the quadratic

We need two numbers that multiply to \(7\times3 = 21\) and add up to \(10\). The numbers are \(7\) and \(3\).
Rewrite the middle term: \(7r^{2}+7r+3r + 3 = 0\)
Group the terms: \((7r^{2}+7r)+(3r + 3)=0\)
Factor out the GCF: \(7r(r + 1)+3(r + 1)=0\)
Factor out \((r + 1)\): \((7r + 3)(r + 1)=0\)

Step 2: Solve for \(r\)

Set each factor equal to zero:
\(7r+3 = 0\) or \(r + 1 = 0\)
For \(7r+3 = 0\): \(7r=-3\Rightarrow r=-\frac{3}{7}\)
For \(r + 1 = 0\): \(r=-1\)

5) Solve \(5v^{2}+16v - 16 = 0\)

Step 1: Factor the quadratic

We need two numbers that multiply to \(5\times(-16)=-80\) and add up to \(16\). The numbers are \(20\) and \(-4\).
Rewrite the middle term: \(5v^{2}+20v-4v - 16 = 0\)
Group the terms: \((5v^{2}+20v)+(-4v - 16)=0\)
Factor out the GCF: \(5v(v + 4)-4(v + 4)=0\)
Factor out \((v + 4)\): \((5v - 4)(v + 4)=0\)

Step 2: Solve for \(v\)

Set each factor equal to zero:
\(5v - 4 = 0\) or \(v + 4 = 0\)
For \(5v - 4 = 0\): \(5v=4\Rightarrow v=\frac{4}{5}\)
For \(v + 4 = 0\): \(v=-4\)

6) Solve \(21a^{2}+41a = 40\)

Step 1: Rewrite in standard form

\(21a^{2}+41a - 40 = 0\)
We need two numbers that multiply to \(21\times(-40)=-840\) and add up to \(41\). The numbers are \(56\) and \(-15\).
Rewrite the middle term: \(21a^{2}+56a-15a - 40 = 0\)
Group the terms: \((21a^{2}+56a)+(-15a - 40)=0\)
Factor out the GCF: \(7a(3a + 8)-5(3a + 8)=0\)
Factor out \((3a + 8)\): \((3a + 8)(7a - 5)=0\)

Step 2: Solve for \(a\)

Set each factor equal to zero:
\(3a+8 = 0\) or \(7a - 5 = 0\)
For \(3a+8 = 0\): \(3a=-8\Rightarrow a=-\frac{8}{3}\)
For \(7a - 5 = 0\): \(7a=5\Rightarrow a=\frac{5}{7}\)

7) Solve \(7v^{2}+19v = 6\)

Step 1: Rewrite in standard form

\(7v^{2}+19v - 6 = 0\)
We need two numbers that multiply to \(7\times(-6)=-42\) and add up to \(19\). The numbers are \(21\) and \(-2\).
Rewrite the middle term: \(7v^{2}+21v-2v - 6 = 0\)
Group the terms: \((7v^{2}+21v)+(-2v - 6)=0\)
Factor out the GCF: \(7v(v + 3)-2(v + 3)=0\)
Factor out \((v + 3)\): \((7v - 2)(v + 3)=0\)

Step 2: Solve for \(v\)

Set each factor equal to zero:
\(7v - 2 = 0\) or \(v + 3 = 0\)
For \(7v - 2 = 0\): \(7v=2\Rightarrow v=\frac{2}{7}\)
For \(v + 3 = 0\): \(v=-3\)

8) Solve \(7b^{2}+21 = 52b\)

Step 1: Rewrite in standard form

\(7b^{2}-52b + 21 = 0\)
We need two numbers that multiply to \(7\times21 = 147\) and add up to \(-52\). The numbers are \(-49\) and \(-3\).
Rewrite the middle term: \(7b^{2}-49b-3b + 21 = 0\)
Group the terms: \((7b^{2}-49b)+(-3b + 21)=0\)
Factor out the GCF: \(7b(b - 7)-3(b - 7)=0\)
Factor out \((b - 7)\): \((7b - 3)(b - 7)=0\)

Step 2: Solve for \(b\)

Set each factor equal to zero:
\(7b - 3 = 0\) or \(b - 7 = 0\)
For \(7b - 3 = 0\): \(7b=3\Rightarrow b=\frac{3}{7}\)
For \(b - 7 = 0\): \(b = 7\)

9) Solve \(3a^{2}=56 - 17a\)

Step 1: Rewrite in standard form

\(3a^{2}+17a - 56 = 0\)
We need two numbers that multiply to \(3\times(-56)=-168\) and add up to \(17\). The numbers are \(24\) and \(-7\).
Rewrite the middle term: \(3a^{2}+24a-7a - 56 = 0\)
Group the terms: \((3a^{2}+24a)+(-7a - 56)=0\)
Factor out the GCF: \(3a(a + 8)-7(a + 8)=0\)
Factor out \((a + 8)\): \((3a - 7)(a + 8)=0\)

Step 2: Solve for \(a\)

Set each factor equal to zero:
\(3a - 7 = 0\) or \(a + 8 = 0\)
For \(3a - 7 = 0\): \(3a=7\Rightarrow a=\frac{7}{3}\)
For \(a + 8 = 0\): \(a=-8\)

10) Solve \(2n^{2}=-n\)

Step 1: Rewrite in standard form

\(2n^{2}+n = 0\)

Step 2: Factor the quadratic

Factor out \(n\): \(n(2n + 1)=0\)

Step 3: Solve for \(n\)

Set each factor equal to zero:
\(n = 0\) or \(2n+1 = 0\)
For \(2n+1 = 0\): \(2n=-1\Rightarrow n=-\frac{1}{2}\)

Answer:

1. \(m = 1, m=-\frac{8}{3}\)
2. \(k=\frac{2}{3}, k=-3\)
3. \(x=-7, x =-\frac{1}{3}\)
4. \(r=-1, r=-\frac{3}{7}\)
5. \(v=\frac{4}{5}, v=-4\)
6. \(a=\frac{5}{7}, a =-\frac{8}{3}\)
7. \(v=\frac{2}{7}, v=-3\)
8. \(b=\frac{3}{7}, b = 7\)
9. \(a=\frac{7}{3}, a=-8\)
10. \(n = 0, n=-\frac{1}{2}\)