4х²-9-11х+19=0
To solve the quadratic equation \( 4x^2 - 11x - 9 + 19 = 0 \), we first simplify it:
- Combine the constants: \[ 4x^2 - 11x + 10 = 0 \]
Next, we can either factor the quadratic or use the quadratic formula. Let's try to factor it first.
We need two numbers that multiply to \( 4 \times 10 = 40 \) and add up to \( -11 \). The numbers that satisfy this are \( -5 \) and \( -6 \).
We can rewrite the middle term: \[ 4x^2 - 5x - 6x + 10 = 0 \]
Now, we group the terms: \[ (4x^2 - 5x) + (-6x + 10) = 0 \]
Factor by grouping: \[ x(4x - 5) - 2(3x - 5) = 0 \]
Factor out the common term: \[ (4x - 5)(x - 2) = 0 \]
Now, set each factor equal to zero:
\( 4x - 5 = 0 \) \[ 4x = 5 \quad \Rightarrow \quad x = \frac{5}{4} \]
\( x - 2 = 0 \) \[ x = 2 \]
So the solutions to the equation \( 4x^2 - 11x + 10 = 0 \) are: \[ x = \frac{5}{4} \quad \text{and} \quad x = 2 \]
