Final answer:
To solve quadratic equations, we can use either the quadratic formula or factor the equation. Depending on the given equation, we choose the most appropriate method. By applying the quadratic formula, we can solve the given equations step-by-step.
Step-by-step explanation:
To solve the equation 3x² + 7x – 6 = 0, we can use the quadratic formula since the equation is not easily factorable. The quadratic formula is x = (-b ± sqrt(b² - 4ac)) / (2a). In this case, a = 3, b = 7, and c = -6. Plugging in these values into the formula, we get x = (-7 ± sqrt(7² - 4(3)(-6))) / (2(3)). Simplifying further, we get x = (-7 ± sqrt(49 + 72)) / 6, which becomes x = (-7 ± sqrt(121)) / 6. Since sqrt(121) = 11, we have x = (-7 ± 11) / 6. This gives us two possible solutions: x = 4/3 and x = -3.
To solve the equation -x² – 2x + 8 = 0, we can again use the quadratic formula since the equation is not easily factorable. In this case, a = -1, b = -2, and c = 8. Plugging in these values into the formula, we get x = (-(-2) ± sqrt((-2)² - 4(-1)(8))) / (2(-1)). Simplifying further, we get x = (2 ± sqrt(4 + 32)) / (-2), which becomes x = (2 ± sqrt(36)) / (-2). Since sqrt(36) = 6, we have x = (2 ± 6) / -2. This gives us two possible solutions: x = -4 and x = 2.
To solve the equation -3x² – 2x – 5 = 0, we can once again use the quadratic formula since the equation is not easily factorable. In this case, a = -3, b = -2, and c = -5. Plugging in these values into the formula, we get x = (-(-2) ± sqrt((-2)² - 4(-3)(-5))) / (2(-3)). Simplifying further, we get x = (2 ± sqrt(4 - 60)) / (-6), which becomes x = (2 ± sqrt(-56)) / (-6). Since sqrt(-56) is not a real number, this equation does not have any real solutions.
To solve the equation x² – 1 = 0, we can factor the equation as the difference of squares. The equation can be written as (x + 1)(x - 1) = 0. Setting each factor equal to zero, we get x + 1 = 0 or x - 1 = 0. Solving for x, we find two possible solutions: x = 1 and x = -1.
To solve the equation x² + 4x – 7 = 0 using the quadratic formula, we can see that this equation is not easily factorable. In this case, a = 1, b = 4, and c = -7. Plugging in these values into the quadratic formula, we get x = (-4 ± sqrt(4² - 4(1)(-7))) / (2(1)). Simplifying further, we get x = (-4 ± sqrt(16 + 28)) / 2, which becomes x = (-4 ± sqrt(44)) / 2. Since sqrt(44) is approximately 6.63, we have x = (-4 ± 6.63) / 2. This gives us two possible solutions: x = 1.315 or x = -5.315.