LT $15$ Abby and Robert are each trying to solve the equation $x^{2}-\sqrt{10 x+26}=0$ . They know that the solutions to $x^{2}=-1$ are $i$ and $-i$ , but they are not sure how to use this information to solve for $x$ in their equation. Solve the equation and explain to them where the $i$ is needed.

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Q. LT

15

Abby and Robert are each trying to solve the equation

x^{2}-\sqrt{10 x+26}=0

. They know that the solutions to

x^{2}=-1

are

i

and

-i

, but they are not sure how to use this information to solve for

x

in their equation. Solve the equation and explain to them where the

i

is needed.

Rewrite Equation: Step $1$ : Rewrite the equation to isolate the square root term. $\newline$ $x^2 - \sqrt{10x + 26} = 0$ $\newline$ $\Rightarrow \sqrt{10x + 26} = x^2$
Square Both Sides: Step $2$ : Square both sides to eliminate the square root. $\newline$ $(\sqrt{10x + 26})^2 = (x^2)^2$ $\newline$ $\Rightarrow 10x + 26 = x^4$
Rearrange Polynomial: Step $3$ : Rearrange the equation to form a standard polynomial equation. $x^4 - 10x - 26 = 0$
Check Rational Roots: Step $4$ : Check for possible rational roots using the Rational Root Theorem. $\newline$ Possible rational roots could be factors of $-26$ divided by factors of $1$ (leading coefficient). $\newline$ Possible roots: $\pm1$ , $\pm2$ , $\pm13$ , $\pm26$
Test Possible Roots: Step $5$ : Test the possible roots in the polynomial. $\newline$ $x = 1$ : $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ (not a root) $\newline$ $x = -1$ : $(-1)^4 - 10\cdot(-1) - 26 = 1 + 10 - 26 = -15$ (not a root) $\newline$ $x = 2$ : $2^4 - 10\cdot2 - 26 = 16 - 20 - 26 = -30$ (not a root) $\newline$ $x = -2$ : $(-2)^4 - 10\cdot(-2) - 26 = 16 + 20 - 26 = 10$ (not a root) $\newline$ $x = 13$ : $13^4 - 10\cdot13 - 26 = 28561 - 130 - 26 = 28405$ (not a root) $\newline$ $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ $0$ : $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ $1$ (not a root) $\newline$ $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ $2$ : $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ $3$ (not a root) $\newline$ $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ $4$ : $1^4 - 10\cdot1 - 26 = 1 - 10 - 26 = -35$ $5$ (not a root)
Consider Complex Roots: Step $6$ : Since no rational roots are found, consider complex roots or use numerical methods or graphing to find roots. $\newline$ This step involves higher-level mathematics or graphing calculators, which might not be readily available.