To solve the quadratic equation [tex]\( 15x^2 - 46x + 16 = 0 \)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 15 \)[/tex], [tex]\( b = -46 \)[/tex], and [tex]\( c = 16 \)[/tex].
1. Calculate the discriminant:
The discriminant ([tex]\( \Delta \)[/tex]) is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
For our equation:
[tex]\[
\Delta = (-46)^2 - 4 \cdot 15 \cdot 16
\][/tex]
Simplify the expression inside the square root:
[tex]\[
\Delta = 2116 - 960 = 1156
\][/tex]
2. Solve for [tex]\( x \)[/tex] using the quadratic formula:
The solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\( b \)[/tex], [tex]\( \Delta \)[/tex], and [tex]\( a \)[/tex] into the formula:
[tex]\[
x = \frac{46 \pm \sqrt{1156}}{30}
\][/tex]
3. Find the square root of the discriminant:
[tex]\[
\sqrt{1156} = 34
\][/tex]
4. Calculate the two possible values for [tex]\( x \)[/tex]:
[tex]\[
x_1 = \frac{46 + 34}{30} = \frac{80}{30} = \frac{8}{3} \approx 2.6666666666666665
\][/tex]
[tex]\[
x_2 = \frac{46 - 34}{30} = \frac{12}{30} = \frac{2}{5} = 0.4
\][/tex]
The solutions to the quadratic equation [tex]\( 15x^2 - 46x + 16 = 0 \)[/tex] are:
[tex]\[
x = \frac{8}{3} \quad \text{and} \quad x = \frac{2}{5}
\][/tex]
Comparing these results with the given options:
A. [tex]\( x = \frac{8}{3}, -\frac{2}{5} \)[/tex]
B. [tex]\( x = \frac{8}{3}, \frac{2}{5} \)[/tex]
C. [tex]\( x = -\frac{8}{3}, -\frac{2}{5} \)[/tex]
D. [tex]\( x = \frac{3}{8}, \frac{2}{5} \)[/tex]
The correct answer is:
B. [tex]\( x = \frac{8}{3}, \frac{2}{5} \)[/tex]
