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$What are the solutions to the equation (2x+1)^(2)-(x+13)=3x^(2)-2x+2 ? Enter your answors in the spaces provided Enter only your answers aos {:[x=],[x=]:}$

What are the solutions to the equation $(2 x+1)^{2}-(x+13)=3 x^{2}-2 x+2$ ? $\newline$ Enter your answors in the spaces provided Enter only your answers $\newline$ aos $\newline$ $\begin{array}{l} x= \\ x= \end{array}$

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Conjugate root theorems

Full solution

Q. What are the solutions to the equation

(2 x+1)^{2}-(x+13)=3 x^{2}-2 x+2

?

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Enter your answors in the spaces provided Enter only your answers

\newline

aos

\newline

\begin{array}{l} x= \\ x= \end{array}

Distribute and Simplify: So we have $4x^2 + 4x + 1 - (x + 13) = 3x^2 - 2x + 2$ . Simplify the left side by distributing the negative sign through $(x + 13)$ .
Combine Like Terms: This gives us $4x^2 + 4x + 1 - x - 13 = 3x^2 - 2x + 2$ . Combine like terms on the left side.
Move Terms and Set to $0$ : We get $4x^2 + 3x - 12 = 3x^2 - 2x + 2$ . Now, let's subtract $3x^2$ , add $2x$ , and subtract $2$ from both sides to move everything to the left side and set the equation to $0$ .
Factor Quadratic Equation: This results in $x^2 + 5x - 14 = 0$ . Now we need to factor this quadratic equation.
Identify Factors: The factors of $-14$ that add up to $5$ are $7$ and $-2$ . So we can write the equation as $(x + 7)(x - 2) = 0$ .
Set Equations Equal: Setting each factor equal to zero gives us the solutions: $x + 7 = 0$ or $x - 2 = 0$ .
Find Solutions: Solving for $x$ , we get $x = -7$ and $x = 2$ . These are the solutions to the equation.

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