The solutions to 2x^2 - 10x + 13 = 0 are a+bi and a-bi, where a and b are positive. What is a*b?
+128448 2x^2 - 10x + 13 = 0
2(x^2 - 5x + 25/4) + 13 - 50/4 = 0
2 ( x - 5/2)^2 + 2/4 = 0
2 ( x - 5/2)^2 = - 1/2 divide both sides by 2
(x - 5/2)^2 = -1/4 take both roots
x - 5/2 = ±1/2 i
x = 5/2 ±1/2 i
a * b = (5/2)(1/2) = 5/4
+26367 The solutions to \(2x^2 - 10x + 13 = 0\) are \(a+bi\) and \(a-bi\), where a and b are positive.
What is \(a*b\)?
\(\begin{array}{|lrcll|} \hline (1) & (a+bi)+(a-bi) &=& \dfrac{10}{2} \\ & (a+bi)+(a-bi) &=& 5 \\ & 2a &=& 5 \\ & \mathbf{a} &=& \mathbf{2.5} \\ \hline (2) & (a+bi)(a-bi) &=& \dfrac{13}{2} \\ & a^2-(bi)^2 &=& 6.5 \\ & a^2-b^2i^2 &=& 6.5 \quad | \quad i^2 = -1 \\ & \mathbf{a^2+b^2} &=& \mathbf{6.5} \quad | \quad a=2.5 \\ & 2.5^2+b^2 &=& 6.5 \\ & b^2 &=& 0.25 \\ & \mathbf{b} &=& \mathbf{0.5} \\ \hline & a*b &=& 2.5*0.5 \\ & \mathbf{a*b} &=& \mathbf{1.25} \\ \hline \end{array}\)
