To find the value of [tex]\( k \)[/tex] that makes the equation [tex]\( k x^2 + k x + 1 = -4 x^2 - x \)[/tex] have equal roots, let's follow these steps:
1. Rewrite the Equation:
First, we need to rewrite the given equation in standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Starting with:
[tex]\[
k x^2 + k x + 1 = -4 x^2 - x
\][/tex]
Move all terms to one side of the equation:
[tex]\[
k x^2 + k x + 1 + 4 x^2 + x = 0
\][/tex]
Combine like terms:
[tex]\[
(k + 4)x^2 + (k + 1)x + 1 = 0
\][/tex]
Now, our equation is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
[tex]\[
a = k + 4, \quad b = k + 1, \quad c = 1
\][/tex]
2. Discriminant Condition for Equal Roots:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have equal roots, its discriminant must be zero. The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = (k + 1)^2 - 4(k + 4) \cdot 1
\][/tex]
3. Simplify the Discriminant:
Let's simplify the expression:
[tex]\[
\Delta = (k + 1)^2 - 4(k + 4)
\][/tex]
[tex]\[
\Delta = (k + 1)^2 - 4k - 16
\][/tex]
[tex]\[
\Delta = k^2 + 2k + 1 - 4k - 16
\][/tex]
[tex]\[
\Delta = k^2 - 2k - 15
\][/tex]
4. Set the Discriminant to Zero and Solve for [tex]\(k\)[/tex]:
For the roots to be equal, the discriminant must be zero:
[tex]\[
k^2 - 2k - 15 = 0
\][/tex]
Solve the quadratic equation for [tex]\(k\)[/tex]:
[tex]\[
k^2 - 2k - 15 = 0
\][/tex]
Factorize the quadratic equation:
[tex]\[
(k - 5)(k + 3) = 0
\][/tex]
Set each factor equal to zero:
[tex]\[
k - 5 = 0 \quad \Rightarrow \quad k = 5
\][/tex]
[tex]\[
k + 3 = 0 \quad \Rightarrow \quad k = -3
\][/tex]
Therefore, the values of [tex]\( k \)[/tex] that make the equation have equal roots are [tex]\(\boxed{-3}\)[/tex] and [tex]\(\boxed{5}\)[/tex].
