ANSWERED] 1 2023 Kuta Software LLC All rights reserved Solving ... - Free Printable

Here are the step-by-step solutions for each quadratic equation.

1) $7x^2 - 3x = 0$
* Step 1: Factor out the common term, which is $x$.
$$x(7x - 3) = 0$$
* Step 2: Set each factor equal to zero and solve.
$$x = 0$$
$$7x - 3 = 0 \rightarrow 7x = 3 \rightarrow x = \frac{3}{7}$$
* Solution: $\left\{ \frac{3}{7}, 0 \right\}$
* Matches Option: B

2) $3b^2 + 20b - 32 = 0$
* Step 1: Find two numbers that multiply to $3 \times -32 = -96$ and add to $20$. These numbers are $24$ and $-4$.
* Step 2: Rewrite the middle term and factor by grouping.
$$3b^2 + 24b - 4b - 32 = 0$$
$$3b(b + 8) - 4(b + 8) = 0$$
$$(3b - 4)(b + 8) = 0$$
* Step 3: Solve for $b$.
$$3b - 4 = 0 \rightarrow b = \frac{4}{3}$$
$$b + 8 = 0 \rightarrow b = -8$$
* Solution: $\left\{ \frac{4}{3}, -8 \right\}$
* Matches Option: A

3) $3p^2 + 14p - 27 = -3$
* Step 1: Set the equation to zero by adding 3 to both sides.
$$3p^2 + 14p - 24 = 0$$
* Step 2: Find two numbers that multiply to $3 \times -24 = -72$ and add to $14$. These numbers are $18$ and $-4$.
* Step 3: Factor by grouping.
$$3p^2 + 18p - 4p - 24 = 0$$
$$3p(p + 6) - 4(p + 6) = 0$$
$$(3p - 4)(p + 6) = 0$$
* Step 4: Solve for $p$.
$$3p - 4 = 0 \rightarrow p = \frac{4}{3}$$
$$p + 6 = 0 \rightarrow p = -6$$
* Solution: $\left\{ \frac{4}{3}, -6 \right\}$
* Matches Option: C

4) $5x^2 + 27x + 31 = 3$
* Step 1: Set the equation to zero by subtracting 3 from both sides.
$$5x^2 + 27x + 28 = 0$$
* Step 2: Find two numbers that multiply to $5 \times 28 = 140$ and add to $27$. These numbers are $20$ and $7$.
* Step 3: Factor by grouping.
$$5x^2 + 20x + 7x + 28 = 0$$
$$5x(x + 4) + 7(x + 4) = 0$$
$$(5x + 7)(x + 4) = 0$$
* Step 4: Solve for $x$.
$$5x + 7 = 0 \rightarrow x = -\frac{7}{5}$$
$$x + 4 = 0 \rightarrow x = -4$$
* Solution: $\left\{ -\frac{7}{5}, -4 \right\}$
* Matches Option: C

5) $7n^2 - 52n = -21$
* Step 1: Set the equation to zero by adding 21 to both sides.
$$7n^2 - 52n + 21 = 0$$
* Step 2: Find two numbers that multiply to $7 \times 21 = 147$ and add to $-52$. These numbers are $-49$ and $-3$.
* Step 3: Factor by grouping.
$$7n^2 - 49n - 3n + 21 = 0$$
$$7n(n - 7) - 3(n - 7) = 0$$
$$(7n - 3)(n - 7) = 0$$
* Step 4: Solve for $n$.
$$7n - 3 = 0 \rightarrow n = \frac{3}{7}$$
$$n - 7 = 0 \rightarrow n = 7$$
* Solution: $\left\{ \frac{3}{7}, 7 \right\}$
* Matches Option: D

6) $2p^2 - 15p = -18$
* Step 1: Set the equation to zero by adding 18 to both sides.
$$2p^2 - 15p + 18 = 0$$
* Step 2: Find two numbers that multiply to $2 \times 18 = 36$ and add to $-15$. These numbers are $-12$ and $-3$.
* Step 3: Factor by grouping.
$$2p^2 - 12p - 3p + 18 = 0$$
$$2p(p - 6) - 3(p - 6) = 0$$
$$(2p - 3)(p - 6) = 0$$
* Step 4: Solve for $p$.
$$2p - 3 = 0 \rightarrow p = \frac{3}{2}$$
$$p - 6 = 0 \rightarrow p = 6$$
* Solution: $\left\{ \frac{3}{2}, 6 \right\}$
* Matches Option: A

7) $3n^2 = 2n$
* Step 1: Move all terms to one side to set it to zero.
$$3n^2 - 2n = 0$$
* Step 2: Factor out the common term $n$.
$$n(3n - 2) = 0$$
* Step 3: Solve for $n$.
$$n = 0$$
$$3n - 2 = 0 \rightarrow 3n = 2 \rightarrow n = \frac{2}{3}$$
* Solution: $\left\{ \frac{2}{3}, 0 \right\}$
* Matches Option: B

8) $9x^2 + 45x - 28 = 2x^2$
* Step 1: Subtract $2x^2$ from both sides to set to zero.
$$7x^2 + 45x - 28 = 0$$
* Step 2: Find two numbers that multiply to $7 \times -28 = -196$ and add to $45$. These numbers are $49$ and $-4$.
* Step 3: Factor by grouping.
$$7x^2 + 49x - 4x - 28 = 0$$
$$7x(x + 7) - 4(x + 7) = 0$$
$$(7x - 4)(x + 7) = 0$$
* Step 4: Solve for $x$.
$$7x - 4 = 0 \rightarrow x = \frac{4}{7}$$
$$x + 7 = 0 \rightarrow x = -7$$
* Solution: $\left\{ \frac{4}{7}, -7 \right\}$
* Matches Option: D

9) $29n^2 - 52n + 32 = 8n^2$
* Step 1: Subtract $8n^2$ from both sides.
$$21n^2 - 52n + 32 = 0$$
* Step 2: Find two numbers that multiply to $21 \times 32 = 672$ and add to $-52$. These numbers are $-28$ and $-24$.
* Step 3: Factor by grouping.
$$21n^2 - 28n - 24n + 32 = 0$$
$$7n(3n - 4) - 8(3n - 4) = 0$$
$$(7n - 8)(3n - 4) = 0$$
* Step 4: Solve for $n$.
$$7n - 8 = 0 \rightarrow n = \frac{8}{7}$$
$$3n - 4 = 0 \rightarrow n = \frac{4}{3}$$
* Solution: $\left\{ \frac{4}{3}, \frac{8}{7} \right\}$
* Matches Option: D

10) $-a^2 - 7a - 25 = 5 - 3a^2$
* Step 1: Move all terms to the left side to set to zero. Add $3a^2$, subtract $5$.
$$(-a^2 + 3a^2) - 7a + (-25 - 5) = 0$$
$$2a^2 - 7a - 30 = 0$$
* Step 2: Find two numbers that multiply to $2 \times -30 = -60$ and add to $-7$. These numbers are $-12$ and $5$.
* Step 3: Factor by grouping.
$$2a^2 - 12a + 5a - 30 = 0$$
$$2a(a - 6) + 5(a - 6) = 0$$
$$(2a + 5)(a - 6) = 0$$
* Step 4: Solve for $a$.
$$2a + 5 = 0 \rightarrow a = -\frac{5}{2}$$
$$a - 6 = 0 \rightarrow a = 6$$
* Solution: $\left\{ -\frac{5}{2}, 6 \right\}$
* Matches Option: D

Final Answer:
1) B
2) A
3) C
4) C
5) D
6) A
7) B
8) D
9) D
10) D