Here are the solutions for each quadratic equation on the worksheet. I have solved them step-by-step to ensure accuracy.
1. $x^2 + 4x + 3 = 0$
* Factor the equation: Find two numbers that multiply to 3 and add to 4. Those numbers are 3 and 1.
* $(x + 3)(x + 1) = 0$
* Set each part to zero: $x + 3 = 0$ or $x + 1 = 0$.
*
Answer: $x = -3, x = -1$
2. $x^2 - 15x + 3 = -51$
* Add 51 to both sides to set to zero: $x^2 - 15x + 54 = 0$.
* Factor: Find two numbers that multiply to 54 and add to -15. Those are -6 and -9.
* $(x - 6)(x - 9) = 0$
*
Answer: $x = 6, x = 9$
3. $x^2 + x - 70 = 2$
* Subtract 2 from both sides: $x^2 + x - 72 = 0$.
* Factor: Find two numbers that multiply to -72 and add to 1. Those are 9 and -8.
* $(x + 9)(x - 8) = 0$
*
Answer: $x = -9, x = 8$
4. $x^2 + 5x - 10 = 14$
* Subtract 14 from both sides: $x^2 + 5x - 24 = 0$.
* Factor: Find two numbers that multiply to -24 and add to 5. Those are 8 and -3.
* $(x + 8)(x - 3) = 0$
*
Answer: $x = -8, x = 3$
5. $x^2 - 8x - 4 = 5$
* Subtract 5 from both sides: $x^2 - 8x - 9 = 0$.
* Factor: Find two numbers that multiply to -9 and add to -8. Those are -9 and 1.
* $(x - 9)(x + 1) = 0$
*
Answer: $x = 9, x = -1$
6. $x^2 - 2x - 35 = 28$
* Subtract 28 from both sides: $x^2 - 2x - 63 = 0$.
* Factor: Find two numbers that multiply to -63 and add to -2. Those are -9 and 7.
* $(x - 9)(x + 7) = 0$
*
Answer: $x = 9, x = -7$
7. $x^2 - 3x = -2$
* Add 2 to both sides: $x^2 - 3x + 2 = 0$.
* Factor: Find two numbers that multiply to 2 and add to -3. Those are -2 and -1.
* $(x - 2)(x - 1) = 0$
*
Answer: $x = 2, x = 1$
8. $x^2 + 4x - 25 = 7$
* Subtract 7 from both sides: $x^2 + 4x - 32 = 0$.
* Factor: Find two numbers that multiply to -32 and add to 4. Those are 8 and -4.
* $(x + 8)(x - 4) = 0$
*
Answer: $x = -8, x = 4$
9. $x^2 + 9x = -8$
* Add 8 to both sides: $x^2 + 9x + 8 = 0$.
* Factor: Find two numbers that multiply to 8 and add to 9. Those are 8 and 1.
* $(x + 8)(x + 1) = 0$
*
Answer: $x = -8, x = -1$
10. $x^2 - 6 = 43$
* Add 6 to both sides: $x^2 = 49$.
* Take the square root of both sides. Remember to include both positive and negative roots.
*
Answer: $x = 7, x = -7$
11. $x^2 - 4x - 1 = 4$
* Subtract 4 from both sides: $x^2 - 4x - 5 = 0$.
* Factor: Find two numbers that multiply to -5 and add to -4. Those are -5 and 1.
* $(x - 5)(x + 1) = 0$
*
Answer: $x = 5, x = -1$
12. $x^2 + 6x - 15 = 12$
* Subtract 12 from both sides: $x^2 + 6x - 27 = 0$.
* Factor: Find two numbers that multiply to -27 and add to 6. Those are 9 and -3.
* $(x + 9)(x - 3) = 0$
*
Answer: $x = -9, x = 3$
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Final Answer:
1. $x = -3, -1$
2. $x = 6, 9$
3. $x = -9, 8$
4. $x = -8, 3$
5. $x = 9, -1$
6. $x = 9, -7$
7. $x = 2, 1$
8. $x = -8, 4$
9. $x = -8, -1$
10. $x = 7, -7$
11. $x = 5, -1$
12. $x = -9, 3$
Parent Tip: Review the logic above to help your child master the concept of algebra practice worksheet.
