To find the solutions of the equation 4x² + 7x – 3 = 24, we first rearrange the equation to set it to zero:
1. Subtract 24 from both sides:
4x² + 7x - 3 - 24 = 0
2. Simplifying this gives:
4x² + 7x - 27 = 0
Now we have a quadratic equation in the standard form ax² + bx + c = 0, where:
- a = 4
- b = 7
- c = -27
Next, we use the quadratic formula to find the solutions:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values of a, b, and c:
x = (-7 ± √(7² - 4 * 4 * -27)) / (2 * 4)
This simplifies to:
x = (-7 ± √(49 + 432)) / 8
Calculating the discriminant:
x = (-7 ± √481) / 8
Now, we find the square root of 481, which is approximately 21.93:
x = (-7 ± 21.93) / 8
This gives us two potential solutions:
- x = (-7 + 21.93) / 8 ≈ 1.74
- x = (-7 – 21.93) / 8 ≈ -3.61
Therefore, the solutions to the equation 4x² + 7x – 3 = 24 are approximately:
- x ≈ 1.74
- x ≈ -3.61
These solutions can be verified by substituting them back into the original equation.