Santiago Lee
07/26/2024 · Senior High School
Use the quadratic formula to solve the quadratic equation \( 2 x^{2}-9 x+11=0 \) Express its solutions in the form \( a \pm b i \) (1 point)
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Step-by-step Solution
Solve the equation \( 2x^2-9x+11=0 \).
Solve the equation(The complex numbers system) by following steps:
- step0: Solve using the quadratic formula in the complex numbers system:
\(2x^{2}-9x+11=0\)
- step1: Solve using the quadratic formula:
\(x=\frac{9\pm \sqrt{\left(-9\right)^{2}-4\times 2\times 11}}{2\times 2}\)
- step2: Simplify the expression:
\(x=\frac{9\pm \sqrt{\left(-9\right)^{2}-4\times 2\times 11}}{4}\)
- step3: Simplify the expression:
\(x=\frac{9\pm \sqrt{-7}}{4}\)
- step4: Simplify the expression:
\(x=\frac{9\pm \sqrt{7}\times i}{4}\)
- step5: Separate into possible cases:
\(\begin{align}&x=\frac{9+\sqrt{7}\times i}{4}\\&x=\frac{9-\sqrt{7}\times i}{4}\end{align}\)
- step6: Simplify the expression:
\(\begin{align}&x=\frac{9}{4}+\frac{\sqrt{7}}{4}i\\&x=\frac{9-\sqrt{7}\times i}{4}\end{align}\)
- step7: Simplify the expression:
\(\begin{align}&x=\frac{9}{4}+\frac{\sqrt{7}}{4}i\\&x=\frac{9}{4}-\frac{\sqrt{7}}{4}i\end{align}\)
- step8: Rewrite:
\(x_{1}=\frac{9}{4}-\frac{\sqrt{7}}{4}i,x_{2}=\frac{9}{4}+\frac{\sqrt{7}}{4}i\)
The solutions to the quadratic equation \(2x^{2}-9x+11=0\) are \(x_{1}=\frac{9}{4}-\frac{\sqrt{7}}{4}i\) and \(x_{2}=\frac{9}{4}+\frac{\sqrt{7}}{4}i\).
Quick Answer
The solutions are \(x_{1}=\frac{9}{4}-\frac{\sqrt{7}}{4}i\) and \(x_{2}=\frac{9}{4}+\frac{\sqrt{7}}{4}i\).
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