Which are the solutions of x² = -11x + 4?
The Correct Answer and Explanation is:
To solve the quadratic equation ( x^2 = -11x + 4 ), we first rearrange it into standard form. This involves moving all terms to one side of the equation:
[
x^2 + 11x – 4 = 0
]
Now, we will apply the quadratic formula, which is given by:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
In our equation, the coefficients are:
- ( a = 1 )
- ( b = 11 )
- ( c = -4 )
Next, we will calculate the discriminant ( b^2 – 4ac ):
[
b^2 – 4ac = 11^2 – 4 \cdot 1 \cdot (-4) = 121 + 16 = 137
]
The discriminant is ( 137 ), which is a positive number. This indicates that there are two distinct real solutions to the equation. Now we can substitute the values into the quadratic formula:
[
x = \frac{-11 \pm \sqrt{137}}{2 \cdot 1} = \frac{-11 \pm \sqrt{137}}{2}
]
Now, let’s calculate the two solutions:
- First solution:
[
x_1 = \frac{-11 + \sqrt{137}}{2}
]
- Second solution:
[
x_2 = \frac{-11 – \sqrt{137}}{2}
]
These two expressions represent the solutions to the equation ( x^2 + 11x – 4 = 0 ).
To summarize, the solutions to the quadratic equation ( x^2 = -11x + 4 ) are:
[
x = \frac{-11 + \sqrt{137}}{2} \quad \text{and} \quad x = \frac{-11 – \sqrt{137}}{2}
]
Conclusion
Using the quadratic formula, we derived two distinct solutions for the equation. The presence of a positive discriminant confirms that both solutions are real and distinct, allowing for effective analysis of the equation’s roots. The solutions can further be evaluated numerically if required for practical applications, but in symbolic form, they provide precise information about the roots of the quadratic equation.