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Assignmax \( x^{2}+3 x+3=0 \)

Ask by Ward Wheeler. in Nigeria
Mar 14,2025

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Answer

The equation \(x^{2}+3x+3=0\) has no real solutions, so the maximum value of \(f(x) = x^{2}+3x+3\) cannot be determined using real numbers.

Solution

Solve the equation \( x^{2}+3x+3=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(x^{2}+3x+3=0\) - step1: Solve using the quadratic formula: \(x=\frac{-3\pm \sqrt{3^{2}-4\times 3}}{2}\) - step2: Simplify the expression: \(x=\frac{-3\pm \sqrt{-3}}{2}\) - step3: Simplify the expression: \(x=\frac{-3\pm \sqrt{3}\times i}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{-3+\sqrt{3}\times i}{2}\\&x=\frac{-3-\sqrt{3}\times i}{2}\end{align}\) - step5: Simplify the expression: \(\begin{align}&x=-\frac{3}{2}+\frac{\sqrt{3}}{2}i\\&x=\frac{-3-\sqrt{3}\times i}{2}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=-\frac{3}{2}+\frac{\sqrt{3}}{2}i\\&x=-\frac{3}{2}-\frac{\sqrt{3}}{2}i\end{align}\) - step7: Rewrite: \(x_{1}=-\frac{3}{2}-\frac{\sqrt{3}}{2}i,x_{2}=-\frac{3}{2}+\frac{\sqrt{3}}{2}i\) - step8: Remove the complex number(s): \(\textrm{No real solution}\) The equation \(x^{2}+3x+3=0\) does not have any real solutions. Therefore, the maximum value of the function \(f(x) = x^{2}+3x+3\) cannot be determined using real numbers.

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Mind Expander

To solve the quadratic equation \( x^2 + 3x + 3 = 0 \), we can use the quadratic formula, which is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 3 \), and \( c = 3 \). Plugging in these values, we first calculate the discriminant \( b^2 - 4ac = 3^2 - 4 \times 1 \times 3 = 9 - 12 = -3 \). Since the discriminant is negative, we will have complex solutions! These can be expressed as \( x = \frac{-3 \pm \sqrt{-3}}{2} \), simplifying to \( x = \frac{-3 \pm i\sqrt{3}}{2} \). Just think about it: those complex solutions mean the graph of the equation doesn’t cross the x-axis at all! Instead, it opens up as a U-shape, floating above the x-axis. Isn’t it intriguing how even in math, a little creativity leads to complex numbers? Plus, knowing this lets you understand more about where certain types of equations can lead you. If you’re itching to dig deeper into quadratics or even polynomials in general, exploring the famous "Completions of Squares" method can be a game changer! It's not only a handy technique for these kinds of equations, but it also emerges in various math realms, from calculus to function analysis. And hey, don't shy away from looking into the fascinating history of how mathematicians grappled with and eventually embraced the concept of imaginary numbers!

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