Solving Quadratic Equations for x with a Coefficients Between -4 ... - Free Printable

Here are the step-by-step solutions for each quadratic equation. To solve these, we generally follow three steps:
1. Set to Zero: Move all numbers to one side so the other side equals $0$.
2. Factor: Break the equation into two parentheses like $(x + a)(x + b) = 0$.
3. Solve: Set each part equal to zero to find the values of $x$.

1. $-2x^2 + 9x + 29 = -6$
* Add $6$ to both sides: $-2x^2 + 9x + 35 = 0$
* Factor the equation: $-(x - 7)(2x + 5) = 0$
* Solve for $x$:
* $x - 7 = 0 \rightarrow x = 7$
* $2x + 5 = 0 \rightarrow 2x = -5 \rightarrow x = -2.5$ (or $-5/2$)

2. $-x^2 - 2x + 7 = -17$
* Add $17$ to both sides: $-x^2 - 2x + 24 = 0$
* Multiply by $-1$ to make it easier: $x^2 + 2x - 24 = 0$
* Factor: $(x + 6)(x - 4) = 0$
* Solve for $x$:
* $x + 6 = 0 \rightarrow x = -6$
* $x - 4 = 0 \rightarrow x = 4$

3. $2x^2 - 3x - 4 = 5$
* Subtract $5$ from both sides: $2x^2 - 3x - 9 = 0$
* Factor: $(x - 3)(2x + 3) = 0$
* Solve for $x$:
* $x - 3 = 0 \rightarrow x = 3$
* $2x + 3 = 0 \rightarrow 2x = -3 \rightarrow x = -1.5$ (or $-3/2$)

4. $4x^2 - 20x + 16 = -8$
* Add $8$ to both sides: $4x^2 - 20x + 24 = 0$
* Divide everything by $4$ to simplify: $x^2 - 5x + 6 = 0$
* Factor: $(x - 2)(x - 3) = 0$
* Solve for $x$:
* $x - 2 = 0 \rightarrow x = 2$
* $x - 3 = 0 \rightarrow x = 3$

5. $2x^2 + 22x + 13 = -23$
* Add $23$ to both sides: $2x^2 + 22x + 36 = 0$
* Divide everything by $2$ to simplify: $x^2 + 11x + 18 = 0$
* Factor: $(x + 9)(x + 2) = 0$
* Solve for $x$:
* $x + 9 = 0 \rightarrow x = -9$
* $x + 2 = 0 \rightarrow x = -2$

6. $-x^2 - 10x - 7 = 2$
* Subtract $2$ from both sides: $-x^2 - 10x - 9 = 0$
* Multiply by $-1$: $x^2 + 10x + 9 = 0$
* Factor: $(x + 1)(x + 9) = 0$
* Solve for $x$:
* $x + 1 = 0 \rightarrow x = -1$
* $x + 9 = 0 \rightarrow x = -9$

7. $x^2 - x - 44 = 12$
* Subtract $12$ from both sides: $x^2 - x - 56 = 0$
* Factor: $(x - 8)(x + 7) = 0$
* Solve for $x$:
* $x - 8 = 0 \rightarrow x = 8$
* $x + 7 = 0 \rightarrow x = -7$

8. $2x^2 - 21x + 28 = -12$
* Add $12$ to both sides: $2x^2 - 21x + 40 = 0$
* Factor: $(2x - 5)(x - 8) = 0$
* Solve for $x$:
* $2x - 5 = 0 \rightarrow 2x = 5 \rightarrow x = 2.5$ (or $5/2$)
* $x - 8 = 0 \rightarrow x = 8$

9. $-2x^2 + 17x - 12 = 9$
* Subtract $9$ from both sides: $-2x^2 + 17x - 21 = 0$
* Multiply by $-1$: $2x^2 - 17x + 21 = 0$
* Factor: $(2x - 3)(x - 7) = 0$
* Solve for $x$:
* $2x - 3 = 0 \rightarrow 2x = 3 \rightarrow x = 1.5$ (or $3/2$)
* $x - 7 = 0 \rightarrow x = 7$

10. $-x^2 - 2x + 19 = -29$
* Add $29$ to both sides: $-x^2 - 2x + 48 = 0$
* Multiply by $-1$: $x^2 + 2x - 48 = 0$
* Factor: $(x + 8)(x - 6) = 0$
* Solve for $x$:
* $x + 8 = 0 \rightarrow x = -8$
* $x - 6 = 0 \rightarrow x = 6$

11. $x^2 + 10x + 4 = -5$
* Add $5$ to both sides: $x^2 + 10x + 9 = 0$
* Factor: $(x + 1)(x + 9) = 0$
* Solve for $x$:
* $x + 1 = 0 \rightarrow x = -1$
* $x + 9 = 0 \rightarrow x = -9$

12. $-4x^2 - 26x - 6 = 30$
* Subtract $30$ from both sides: $-4x^2 - 26x - 36 = 0$
* Divide by $-2$ to simplify: $2x^2 + 13x + 18 = 0$
* Factor: $(2x + 9)(x + 2) = 0$
* Solve for $x$:
* $2x + 9 = 0 \rightarrow 2x = -9 \rightarrow x = -4.5$ (or $-9/2$)
* $x + 2 = 0 \rightarrow x = -2$

Final Answer:
1. $x = 7, -2.5$
2. $x = 4, -6$
3. $x = 3, -1.5$
4. $x = 2, 3$
5. $x = -9, -2$
6. $x = -1, -9$
7. $x = 8, -7$
8. $x = 2.5, 8$
9. $x = 1.5, 7$
10. $x = -8, 6$
11. $x = -1, -9$
12. $x = -2, -4.5$