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Solve each equation using the quadratic formula.

2) [tex]\(5n^2 - 4n - 3 = -10\)[/tex]

Sagot :

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.
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