2x2=x+21=0

To solve the equation \( 2x^2 = x + 21 \), we first need to rearrange it into standard quadratic form:

1. Start with the equation: \[ 2x^2 - x - 21 = 0 \]

2. Now, we can solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = -1 \), and \( c = -21 \).

3. Calculate the discriminant \( b^2 - 4ac \): \[ (-1)^2 - 4(2)(-21) = 1 + 168 = 169 \]

4. Substitute the values into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{169}}{2 \cdot 2} = \frac{1 \pm 13}{4} \]

5. Calculate the two possible values for \( x \):

   • For \( x = \frac{1 + 13}{4} = \frac{14}{4} = \frac{7}{2} = 3.5 \)
   • For \( x = \frac{1 - 13}{4} = \frac{-12}{4} = -3 \)

Thus, the solutions to the equation \( 2x^2 = x + 21 \) are: \[ x = 3.5 \quad \text{and} \quad x = -3 \]