Find the value of x in the equation 4x^2 + 8x + 3 = 0.

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Question: Find the value of x in the equation 4x^2 + 8x + 3 = 0.

Options:

Correct Answer: x = -3/2

Solution:

Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a gives x = [-8 ± √(64 - 48)] / 8 = [-8 ± 4] / 8.

Find the value of x in the equation 4x^2 + 8x + 3 = 0.

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.