What are the solutions to the equation 4x^2 + 7x – 3 = 24?

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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:

  1. x = (-7 + 21.93) / 8 ≈ 1.74
  2. 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.

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