To find the value of \( k \) for which the quadratic equation \((k-1)x^2 + (k+4)x + (k+7) = 0\) has equal roots, we can follow these steps
Now Understand the Condition for Equal Roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \) to have equal roots, the **discriminant** must be zero. The discriminant \(\Delta\) is given by:
\[\Delta = b^2 - 4ac\]
If \(\Delta = 0\), the roots are equal.
Now Identify \( a \), \( b \), and \( c \)
In the equation \((k-1)x^2 + (k+4)x + (k+7) = 0\), we can identify:
- \( a = (k-1) \)
- \( b = (k+4) \)
- \( c = (k+7) \)
Now Write the Discriminant Formula
Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
\[\Delta = (k+4)^2 - 4(k-1)(k+7)\]
Now Simplify the Expression
Now, we'll expand and simplify the expression:
1. Expand \( (k+4)^2 \):
\[ (k+4)^2 = k^2 + 8k + 16\]
2. Expand \( 4(k-1)(k+7) \):
\[4(k-1)(k+7) = 4[k^2 + 7k - k - 7] = 4(k^2 + 6k - 7) = 4k^2 + 24k - 28 \]
Now, put everything together:
\[\Delta = k^2 + 8k + 16 - (4k^2 + 24k - 28)\]
Now Simplify Further
Combine like terms:
\[\Delta = k^2 + 8k + 16 - 4k^2 - 24k + 28\]
\[\Delta = -3k^2 - 16k + 44\]
Now Set the Discriminant to Zero
Since we want the roots to be equal, set the discriminant to zero:
\[-3k^2 - 16k + 44 = 0\]
Now Solve the Quadratic Equation
This is now a quadratic equation in \( k \). We can solve it using the quadratic formula:
\[k = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(-3)(44)}}{2(-3)}\]
Simplify:
\[k = \frac{16 \pm \sqrt{256 + 528}}{-6}\]
\[k = \frac{16 \pm \sqrt{784}}{-6}\]
\[k = \frac{16 \pm 28}{-6}\]
This gives us two solutions:
1. \( k = \frac{16 + 28}{-6} = \frac{44}{-6} = -\frac{22}{3} \)
2. \( k = \frac{16 - 28}{-6} = \frac{-12}{-6} = 2 \)
The values of \( k \) for which the equation has equal roots are \( k = 2 \) and \( k = -\frac{22}{3} \).