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The equation \(9m^2 - 11m + 18 = 0\) has no real solutions.
Solution
Solve the equation \( 9m^2 - 11m + 18 = 0 \).
Solve the equation(The complex numbers system) by following steps:
- step0: Solve using the quadratic formula in the complex numbers system:
\(9m^{2}-11m+18=0\)
- step1: Solve using the quadratic formula:
\(m=\frac{11\pm \sqrt{\left(-11\right)^{2}-4\times 9\times 18}}{2\times 9}\)
- step2: Simplify the expression:
\(m=\frac{11\pm \sqrt{\left(-11\right)^{2}-4\times 9\times 18}}{18}\)
- step3: Simplify the expression:
\(m=\frac{11\pm \sqrt{-527}}{18}\)
- step4: Simplify the expression:
\(m=\frac{11\pm \sqrt{527}\times i}{18}\)
- step5: Separate into possible cases:
\(\begin{align}&m=\frac{11+\sqrt{527}\times i}{18}\\&m=\frac{11-\sqrt{527}\times i}{18}\end{align}\)
- step6: Simplify the expression:
\(\begin{align}&m=\frac{11}{18}+\frac{\sqrt{527}}{18}i\\&m=\frac{11-\sqrt{527}\times i}{18}\end{align}\)
- step7: Simplify the expression:
\(\begin{align}&m=\frac{11}{18}+\frac{\sqrt{527}}{18}i\\&m=\frac{11}{18}-\frac{\sqrt{527}}{18}i\end{align}\)
- step8: Rewrite:
\(m_{1}=\frac{11}{18}-\frac{\sqrt{527}}{18}i,m_{2}=\frac{11}{18}+\frac{\sqrt{527}}{18}i\)
- step9: Remove the complex number(s):
\(\textrm{No real solution}\)
The given quadratic equation \(9m^2 - 11m + 18 = 0\) does not have any real solutions. This means that the equation has no real roots.
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Simplify this solution Extra Insights
Did you know that quadratic equations like \( 9m^2 - 11m + 18 = 0 \) can be solved using the quadratic formula? This formula is \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a, b, \) and \( c \) are the coefficients from \( am^2 + bm + c = 0 \). In your case, \( a = 9, b = -11, \) and \( c = 18 \). So, just plug those values in to find your solutions! When solving quadratic equations, one common mistake is miscalculating the discriminant (\( b^2 - 4ac \)). Make sure to check your arithmetic carefully! A negative discriminant means there are no real solutions, which can be a head-scratcher. Now you’re ready to tackle it confidently!
