Question

例）$x^2 = 4x + 24$

Explanation:

Step1: Rearrange the equation

We start with the equation \( k^{2}=4k + 24 \). To solve for \( k \), we first rearrange it into a quadratic equation in standard form \( ax^{2}+bx + c = 0 \). Subtract \( 4k \) and \( 24 \) from both sides:
\( k^{2}-4k - 24=0 \)

Step2: Use the quadratic formula

For a quadratic equation \( ax^{2}+bx + c = 0 \), the quadratic formula is \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \). In our equation \( k^{2}-4k - 24=0 \), we have \( a = 1 \), \( b=-4 \), and \( c=-24 \).

First, calculate the discriminant \( D=b^{2}-4ac \):
\( D=(-4)^{2}-4\times1\times(-24) \)
\( D = 16 + 96 \)
\( D=112 \)

Then, find the square root of the discriminant: \( \sqrt{112}=\sqrt{16\times7}=4\sqrt{7} \)

Now, apply the quadratic formula:
\( k=\frac{-(-4)\pm4\sqrt{7}}{2\times1}=\frac{4\pm4\sqrt{7}}{2}=2\pm2\sqrt{7} \)

Answer:

\( k = 2 + 2\sqrt{7} \) or \( k = 2 - 2\sqrt{7} \)