To solve the quadratic equation [tex]\( 5x^2 - 4x = 6 \)[/tex], we need to put it in standard form: [tex]\( ax^2 + bx + c = 0 \)[/tex].
First, subtract 6 from both sides:
[tex]\[ 5x^2 - 4x - 6 = 0. \][/tex]
Now, we identify the coefficients:
[tex]\[ a = 5, \quad b = -4, \quad c = -6. \][/tex]
Next, we calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac. \][/tex]
Plugging in the values:
[tex]\[ \Delta = (-4)^2 - 4(5)(-6). \][/tex]
[tex]\[ \Delta = 16 + 120. \][/tex]
[tex]\[ \Delta = 136. \][/tex]
The discriminant ([tex]\(\Delta\)[/tex]) is 136.
Using the quadratic formula to find the roots:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a}. \][/tex]
We substitute in the values:
[tex]\[ x = \frac{-(-4) \pm \sqrt{136}}{2 \times 5}. \][/tex]
Simplify:
[tex]\[ x = \frac{4 \pm \sqrt{136}}{10}. \][/tex]
Therefore, the solutions to the equation are:
[tex]\[ x = \frac{4 + \sqrt{136}}{10} \quad \text{and} \quad x = \frac{4 - \sqrt{136}}{10}. \][/tex]
For this specific problem, since we need to match this result to one of the option choices, let’s analyze the answers provided:
- A. [tex]\( x = \frac{-2 \pm \sqrt{26}}{5} \)[/tex]
- B. [tex]\( x = \frac{-2 \pm \sqrt{34}}{5} \)[/tex]
- C. [tex]\( x = \frac{2 \pm \sqrt{26}}{5} \)[/tex]
- D. [tex]\( x = \frac{2 \pm \sqrt{34}}{5} \)[/tex]
To match, let's express our solution:
[tex]\[ x = \frac{4 \pm \sqrt{136}}{10}. \][/tex]
4 divided by 10 is:
[tex]\[ \frac{4}{10} = 0.4. \][/tex]
Therefore,
[tex]\[ x = \frac{0.4 \pm \sqrt{136}}{2}. \][/tex]
It’s more straightforward to compare directly to our standard roots found:
[tex]\[ x = \frac{2 \pm \sqrt{34}}{5}.\][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\frac{2 \pm \sqrt{34}}{5}}. \][/tex]
Therefore, the correct answer is:
D. [tex]\( x = \frac{2 \pm \sqrt{34}}{5} \)[/tex]
