Rounded to the nearest hundredth, what is the positive solution to the quadratic equation [tex]\(0 = 2x^2 + 3x - 8\)[/tex]?

Quadratic formula: [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]

A. 1.39
B. 2.00
C. 2.89
D. 3.50



Answer :

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]