To solve the quadratic equation [tex]\(5n^2 - 4n - 3 = -10\)[/tex], follow these steps:
1. Move all terms to one side to set the equation to zero.
[tex]\[5n^2 - 4n - 3 + 10 = 0\][/tex]
[tex]\[5n^2 - 4n + 7 = 0\][/tex]
Now, we have the quadratic equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
2. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the equation [tex]\(5n^2 - 4n + 7 = 0\)[/tex].
[tex]\[a = 5\][/tex]
[tex]\[b = -4\][/tex]
[tex]\[c = 7\][/tex]
3. Calculate the discriminant [tex]\(\Delta\)[/tex] using the formula [tex]\(\Delta = b^2 - 4ac\)[/tex].
[tex]\[
\Delta = (-4)^2 - 4(5)(7)
\][/tex]
[tex]\[
\Delta = 16 - 140
\][/tex]
[tex]\[
\Delta = -124
\][/tex]
4. Analyze the discriminant:
If [tex]\(\Delta < 0\)[/tex], the quadratic equation has no real solutions.
Here, [tex]\(\Delta = -124\)[/tex], which is less than 0.
Therefore, the quadratic equation [tex]\(5n^2 - 4n + 7 = 0\)[/tex] has no real solutions.
