To solve the quadratic equation [tex]\(0 = 2x^2 + 3x - 8\)[/tex] and find the positive solution rounded to the nearest hundredth, we'll use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
[tex]\[ a = 2, \quad b = 3, \quad c = -8 \][/tex]
1. Calculate the Discriminant:
The discriminant of a quadratic equation is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
[tex]\[
\Delta = 3^2 - 4 \cdot 2 \cdot (-8) = 9 + 64 = 73
\][/tex]
2. Calculate the Square Root of the Discriminant:
[tex]\[
\sqrt{73} \approx 8.544
\][/tex]
3. Apply the Quadratic Formula:
Substitute the values into the quadratic formula to find the two solutions:
[tex]\[
x_{1} = \frac{-3 + \sqrt{73}}{2 \cdot 2} = \frac{-3 + 8.544}{4} = \frac{5.544}{4} = 1.386
\][/tex]
[tex]\[
x_{2} = \frac{-3 - \sqrt{73}}{2 \cdot 2} = \frac{-3 - 8.544}{4} = \frac{-11.544}{4} = -2.886
\][/tex]
4. Identify the Positive Solution:
Among the two solutions, the positive solution is [tex]\( x = 1.386 \)[/tex].
5. Round to the Nearest Hundredth:
Rounding [tex]\( 1.386 \)[/tex] to the nearest hundredth, we get [tex]\( 1.39 \)[/tex].
Therefore, the positive solution to the quadratic equation [tex]\(0 = 2x^2 + 3x - 8\)[/tex], rounded to the nearest hundredth, is:
[tex]\[ \boxed{1.39} \][/tex]
