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# Help

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The solutions to 2x^2 - 10x + 13 = 0 are a+bi and a-bi, where a and b are positive. What is a*b?

Apr 7, 2020

#1
+114141
+1

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

Apr 7, 2020
#2
+25648
+2

The solutions to $$2x^2 - 10x + 13 = 0$$ are $$a+bi$$ and $$a-bi$$, where a and b are positive.
What is $$a*b$$?

$$\begin{array}{|lrcll|} \hline (1) & (a+bi)+(a-bi) &=& \dfrac{10}{2} \\ & (a+bi)+(a-bi) &=& 5 \\ & 2a &=& 5 \\ & \mathbf{a} &=& \mathbf{2.5} \\ \hline (2) & (a+bi)(a-bi) &=& \dfrac{13}{2} \\ & a^2-(bi)^2 &=& 6.5 \\ & a^2-b^2i^2 &=& 6.5 \quad | \quad i^2 = -1 \\ & \mathbf{a^2+b^2} &=& \mathbf{6.5} \quad | \quad a=2.5 \\ & 2.5^2+b^2 &=& 6.5 \\ & b^2 &=& 0.25 \\ & \mathbf{b} &=& \mathbf{0.5} \\ \hline & a*b &=& 2.5*0.5 \\ & \mathbf{a*b} &=& \mathbf{1.25} \\ \hline \end{array}$$

Apr 7, 2020