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Find the sum of the squares of the solutions of x^2-13x+4=5x+7.

Apr 17, 2022

#1
+9459
+2

Move every term to one side:

$$\begin{array}{rcl} x^2 -13x + 4&=& 5x + 7\\ x^2 - 18x - 3 &=& 0 \end{array}$$

Let a and b be the roots. Then by Vieta's formulae,

$$\begin{cases}a + b =18\\ab=-3\end{cases}$$

Therefore,

$$\begin{array}{cl} &\text{sum of square of roots}\\ =& a^2 + b^2\\ =& (a + b)^2 - 2ab\\ =& 18^2 - 2(-3)\\ =& 330 \end{array}$$

Apr 17, 2022

#1
+9459
+2

Move every term to one side:

$$\begin{array}{rcl} x^2 -13x + 4&=& 5x + 7\\ x^2 - 18x - 3 &=& 0 \end{array}$$

Let a and b be the roots. Then by Vieta's formulae,

$$\begin{cases}a + b =18\\ab=-3\end{cases}$$

Therefore,

$$\begin{array}{cl} &\text{sum of square of roots}\\ =& a^2 + b^2\\ =& (a + b)^2 - 2ab\\ =& 18^2 - 2(-3)\\ =& 330 \end{array}$$

MaxWong Apr 17, 2022