To find the approximate solutions of the equation 4x² + 2x – 17 = 0, we can use the quadratic formula given by:
x = (-b ± √(b² – 4ac)) / 2a
In our case, the coefficients are:
- a = 4
- b = 2
- c = -17
First, let’s calculate the discriminant:
b² – 4ac = (2)² – 4(4)(-17)
= 4 + 272 = 276
Next, we will substitute the values into the quadratic formula:
x = (-2 ± √276) / (2 * 4)
The square root of 276 can be simplified:
√276 = √(4 * 69) = 2√69
Now, substituting that back into our formula gives:
x = (-2 ± 2√69) / 8
Which can be simplified further:
x = -1/4 ± √69 / 4
Now calculating the two solutions:
Solution 1:
x = -1/4 + √69/4
Using a calculator, √69 ≈ 8.3066, thus:
x ≈ -0.25 + 2.0765 ≈ 1.8265
Rounded to the nearest hundredth, this gives x ≈ 1.83.
Solution 2:
x = -1/4 – √69/4
Using the approximation for √69 again:
x ≈ -0.25 – 2.0765 ≈ -2.3265
Rounded to the nearest hundredth, this gives x ≈ -2.33.
In summary, the approximate solutions for the equation 4x² + 2x – 17 = 0, rounded to the nearest hundredth, are:
- x ≈ 1.83
- x ≈ -2.33