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Solve the equation for all real solutions in simplest form. $\newline$ $2a^{2}-5a-2=-3a^{2}$

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Quadratic equation with complex roots

Full solution

Q. Solve the equation for all real solutions in simplest form.

\newline

2a^{2}-5a-2=-3a^{2}

Move Terms to One Side: Move all terms to one side of the equation to set it equal to zero. $\newline$ Add $3a^2$ to both sides of the equation to get all the $a^2$ terms on one side. $\newline$ $2a^2 - 5a - 2 + 3a^2 = -3a^2 + 3a^2$ $\newline$ $5a^2 - 5a - 2 = 0$
Factor the Quadratic Equation: Factor the quadratic equation if possible. $\newline$ We need to find two numbers that multiply to $5*(-2) = -10$ and add up to $-5$ . These numbers are $-5$ and $+2$ . $\newline$ $(5a + 2)(a - 1) = 0$
Set Factors Equal: Set each factor equal to zero and solve for $a$ . $5a + 2 = 0$ or $a - 1 = 0$ For the first factor: $5a + 2 = 0$ $5a = -2$ $a = -\frac{2}{5}$ For the second factor: $a - 1 = 0$ $a = 1$
Write Solutions: Write the solutions in simplest form. $\newline$ The solutions are $a = -\frac{2}{5}$ and $a = 1$ .

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