To solve the equation 4x² + 3 = 12x, we first need to rearrange it into a standard quadratic form. This means we set the equation to zero:
4x² – 12x + 3 = 0
Next, we can use the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, a = 4, b = -12, and c = 3. We will substitute these values into the formula:
Step 1: Calculate the discriminant (b² – 4ac):
b² = (-12)² = 144
4ac = 4 * 4 * 3 = 48
Discriminant = 144 – 48 = 96
Step 2: Apply the quadratic formula:
x = (12 ± √96) / (2 * 4)
Step 3: Simplify √96:
√96 = √(16 * 6) = 4√6
So, substituting back:
x = (12 ± 4√6) / 8
Step 4: Simplify the expression further:
x = (12/8) ± (4√6/8)
x = 1.5 ± (√6 / 2)
Now we can calculate the approximate values:
√6 is approximately 2.45, so:
x = 1.5 ± (2.45 / 2)
x = 1.5 ± 1.225
This gives us two solutions:
x₁ = 1.5 + 1.225 ≈ 2.725
x₂ = 1.5 – 1.225 ≈ 0.275
Final Step: Round to the nearest hundredth:
Thus, the approximate solutions to the nearest hundredth are:
x₁ ≈ 2.73 and x₂ ≈ 0.28