Find the value of x in the equation 4x^2 + 8x + 3 = 0.
Correct Answer: x = -0.5 or x = -0.75
- Step 1: Identify the coefficients in the equation 4x^2 + 8x + 3 = 0. Here, a = 4, b = 8, and c = 3.
- Step 2: Write down the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.
- Step 3: Calculate b²: 8² = 64.
- Step 4: Calculate 4ac: 4 * 4 * 3 = 48.
- Step 5: Calculate the discriminant (b² - 4ac): 64 - 48 = 16.
- Step 6: Find the square root of the discriminant: √16 = 4.
- Step 7: Substitute b and the square root of the discriminant into the formula: x = [-8 ± 4] / (2 * 4).
- Step 8: Calculate the denominator: 2 * 4 = 8.
- Step 9: Solve for the two possible values of x: x = (-8 + 4) / 8 and x = (-8 - 4) / 8.
- Step 10: Calculate the first value: (-8 + 4) / 8 = -4 / 8 = -0.5.
- Step 11: Calculate the second value: (-8 - 4) / 8 = -12 / 8 = -1.5.
- Step 12: The two solutions for x are -0.5 and -1.5.
- Quadratic Equations – The question tests the ability to solve a quadratic equation using the quadratic formula.
- Discriminant – Understanding the discriminant (b² - 4ac) is crucial for determining the nature of the roots.
- Algebraic Manipulation – The question requires correct manipulation of algebraic expressions to arrive at the solution.
