Quadratic equations practice worksheet for algebra students.

Here are the step-by-step solutions for each quadratic equation on the worksheet. I will solve them by factoring, which is a common method for these types of problems.

1. $2x^2 - 2x - 24 = 0$

Step 1: Notice that all numbers (2, -2, -24) are divisible by 2. Divide the entire equation by 2 to make it simpler.
$$x^2 - x - 12 = 0$$

Step 2: Find two numbers that multiply to -12 and add up to -1.
* Factors of -12: (-4, 3), (4, -3), (-6, 2), etc.
* Check sums: $-4 + 3 = -1$. This works.

Step 3: Write the factors.
$$(x - 4)(x + 3) = 0$$

Step 4: Solve for $x$.
$x - 4 = 0 \rightarrow x = 4$
$x + 3 = 0 \rightarrow x = -3$

Answer: $x = 4, x = -3$

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2. $x^2 + 10x + 16 = 0$

Step 1: Find two numbers that multiply to 16 and add up to 10.
* Factors of 16: (1, 16), (2, 8), (4, 4).
* Check sums: $2 + 8 = 10$. This works.

Step 2: Write the factors.
$$(x + 2)(x + 8) = 0$$

Step 3: Solve for $x$.
$x + 2 = 0 \rightarrow x = -2$
$x + 8 = 0 \rightarrow x = -8$

Answer: $x = -2, x = -8$

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3. $4x^2 + 15x + 9 = 0$

Step 1: Since the first number is not 1, we use the "ac method" or trial and error. We need two numbers that multiply to $4 \times 9 = 36$ and add to 15.
* Factors of 36: (1, 36), (2, 18), (3, 12)...
* Check sums: $3 + 12 = 15$. This works.

Step 2: Split the middle term ($15x$) using 3 and 12.
$$4x^2 + 12x + 3x + 9 = 0$$

Step 3: Factor by grouping.
Group the first two and last two terms:
$4x(x + 3) + 3(x + 3) = 0$

Step 4: Combine the groups.
$$(4x + 3)(x + 3) = 0$$

Step 5: Solve for $x$.
$4x + 3 = 0 \rightarrow 4x = -3 \rightarrow x = -\frac{3}{4}$
$x + 3 = 0 \rightarrow x = -3$

Answer: $x = -\frac{3}{4}, x = -3$

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4. $x^2 - 13x + 42 = 0$

Step 1: Find two numbers that multiply to 42 and add up to -13.
* Since the product is positive and the sum is negative, both numbers must be negative.
* Factors of 42: (-1, -42), (-2, -21), (-6, -7)...
* Check sums: $-6 + (-7) = -13$. This works.

Step 2: Write the factors.
$$(x - 6)(x - 7) = 0$$

Step 3: Solve for $x$.
$x - 6 = 0 \rightarrow x = 6$
$x - 7 = 0 \rightarrow x = 7$

Answer: $x = 6, x = 7$

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5. $6x^2 - x - 40 = 0$

Step 1: Find two numbers that multiply to $6 \times -40 = -240$ and add to -1.
* We need factors of 240 that are very close to each other (since they add to -1).
* $\sqrt{240}$ is roughly 15. Let's try numbers near 15.
* $15 \times 16 = 240$.
* To get -1, we use -16 and +15.

Step 2: Split the middle term ($-x$) using -16 and 15.
$$6x^2 - 16x + 15x - 40 = 0$$

Step 3: Factor by grouping.
Group first two: $2x(3x - 8)$
Group last two: $+5(3x - 8)$

Step 4: Combine the groups.
$$(2x + 5)(3x - 8) = 0$$

Step 5: Solve for $x$.
$2x + 5 = 0 \rightarrow 2x = -5 \rightarrow x = -\frac{5}{2}$ (or -2.5)
$3x - 8 = 0 \rightarrow 3x = 8 \rightarrow x = \frac{8}{3}$

Answer: $x = -\frac{5}{2}, x = \frac{8}{3}$

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6. $2x^2 + 13x + 15 = 0$

Step 1: Find two numbers that multiply to $2 \times 15 = 30$ and add to 13.
* Factors of 30: (1, 30), (2, 15), (3, 10)...
* Check sums: $3 + 10 = 13$. This works.

Step 2: Split the middle term ($13x$) using 3 and 10.
$$2x^2 + 10x + 3x + 15 = 0$$

Step 3: Factor by grouping.
Group first two: $2x(x + 5)$
Group last two: $+3(x + 5)$

Step 4: Combine the groups.
$$(2x + 3)(x + 5) = 0$$

Step 5: Solve for $x$.
$2x + 3 = 0 \rightarrow 2x = -3 \rightarrow x = -\frac{3}{2}$ (or -1.5)
$x + 5 = 0 \rightarrow x = -5$

Answer: $x = -\frac{3}{2}, x = -5$

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Final Answer:
1. $x = 4, x = -3$
2. $x = -2, x = -8$
3. $x = -\frac{3}{4}, x = -3$
4. $x = 6, x = 7$
5. $x = -\frac{5}{2}, x = \frac{8}{3}$
6. $x = -\frac{3}{2}, x = -5$