299х2+100х=500-101х2

Let's solve the equation step by step:

Given equation: \[ 299x^2 + 100x = 500 - 101x^2 \]

First, let's move all terms to one side of the equation: \[ 299x^2 + 100x + 101x^2 - 500 = 0 \]

Combine like terms: \[ (299x^2 + 101x^2) + 100x - 500 = 0 \] \[ 400x^2 + 100x - 500 = 0 \]

Now, we have a standard quadratic equation of the form: \[ ax^2 + bx + c = 0 \] where \( a = 400 \), \( b = 100 \), and \( c = -500 \).

We can simplify this equation by dividing all terms by 100: \[ 4x^2 + x - 5 = 0 \]

Now, we can solve for \( x \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting in the values of \( a \), \( b \), and \( c \): \[ x = \frac{-1 \pm \sqrt{(1)^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4} \] \[ x = \frac{-1 \pm \sqrt{1 + 80}}{8} \] \[ x = \frac{-1 \pm \sqrt{81}}{8} \] \[ x = \frac{-1 \pm 9}{8} \]

Now, we calculate the two possible values for \( x \):

1. \( x = \frac{-1 + 9}{8} = \frac{8}{8} = 1 \)
2. \( x = \frac{-1 - 9}{8} = \frac{-10}{8} = -\frac{5}{4} \)

Thus, the solutions to the equation \( 299x^2 + 100x = 500 - 101x^2 \) are: \[ x = 1 \quad \text{and} \quad x = -\frac{5}{4} \]