To solve the quadratic equation [tex]\(2n^2 = 10n + 48\)[/tex], follow these steps:
1. Rewrite the equation in standard form:
A quadratic equation should be written in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. Start by moving all terms to one side of the equation to set it to zero.
[tex]\[
2n^2 - 10n - 48 = 0
\][/tex]
2. Simplify the equation:
The current equation [tex]\(2n^2 - 10n - 48 = 0\)[/tex] is already simplified with all terms on one side.
3. Factor the quadratic equation:
We look to factorize the quadratic equation [tex]\(2n^2 - 10n - 48\)[/tex]. To do this, let's find numbers that multiply to [tex]\(2 \cdot (-48) = -96\)[/tex] and add up to [tex]\(-10\)[/tex].
The pair of numbers that meet this condition are [tex]\(-16\)[/tex] and [tex]\(6\)[/tex], because:
[tex]\[
-16 \cdot 6 = -96 \quad \text{and} \quad -16 + 6 = -10
\][/tex]
4. Rewrite the middle term using these numbers:
Split the middle term [tex]\(-10n\)[/tex] using [tex]\(-16n\)[/tex] and [tex]\(6n\)[/tex].
[tex]\[
2n^2 - 16n + 6n - 48 = 0
\][/tex]
5. Factor by grouping:
Group the terms to factor by grouping:
[tex]\[
(2n^2 - 16n) + (6n - 48) = 0
\][/tex]
Factor out the common factors from each group:
[tex]\[
2n(n - 8) + 6(n - 8) = 0
\][/tex]
6. Factor out the common binomial factor:
Notice that [tex]\((n - 8)\)[/tex] is common in both groups.
[tex]\[
(2n + 6)(n - 8) = 0
\][/tex]
Simplify further:
[tex]\[
2(n + 3)(n - 8) = 0
\][/tex]
7. Set each factor to zero and solve for [tex]\(n\)[/tex]:
[tex]\[
2(n + 3) = 0 \quad \text{or} \quad n - 8 = 0
\][/tex]
Solve for [tex]\(n\)[/tex] in each case:
[tex]\[
n + 3 = 0 \implies n = -3
\][/tex]
[tex]\[
n - 8 = 0 \implies n = 8
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(2n^2 = 10n + 48\)[/tex] are:
[tex]\[
\boxed{-3 \text{ and } 8}
\][/tex]
