Abby and Robert are each trying to solve the equation x^(2)-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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Rewrite Equation:  Step 1: Rewrite the equation to isolate the square root term. x^2 - sqrt(10x + 26) = 0 => sqrt(10x + 26) = x^2 Square Both Sides:  Step 2: Square both sides to eliminate the square root. (sqrt(10x + 26))^2 = (x^2)^2 => 10x + 26 = x^4 Rearrange Polynomial:  Step 3: Rearrange the equation to form a standard polynomial equation. x^4 - 10x - 26 = 0 Factorize or Find Roots:  Step 4: Attempt to factorize the polynomial or use a numerical method to find roots. This polynomial is not easily factorizable, so numerical methods or graphing might be needed to find the roots. Check Imaginary Roots:  Step 5: Check if any roots are imaginary. Since the original equation involved squaring both sides, we need to check if any solutions involve imaginary numbers. However, the polynomial x^4 - 10x - 26 = 0 does not directly suggest imaginary roots without further analysis or numerical solution.

Abby and Robert are each trying to solve the equation $x^{2}-10x+26=0$ . 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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