Question - Solving Quadratic Equations with Quadratic Formula
Solution:
To solve the quadratic equation $$ a(a - 3) = -1 $$, you need to first expand the equation and then solve for $$ a $$.Step 1: Expand the left side of the equation:$$ a \cdot a = a^2 $$$$ a \cdot (-3) = -3a $$So, the expanded form is:$$ a^2 - 3a $$Step 2: Bring all terms to one side to set the quadratic equation equal to zero:$$ a^2 - 3a + 1 = 0 $$Step 3: Solve the quadratic equation. This can be done using the quadratic formula $$ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where in our equation, $$ a = 1 $$, $$ b = -3 $$, and $$ c = 1 $$.The quadratic formula gives:$$ a = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} $$$$ a = \frac{3 \pm \sqrt{9 - 4}}{2} $$$$ a = \frac{3 \pm \sqrt{5}}{2} $$So the solutions are:$$ a = \frac{3 + \sqrt{5}}{2} $$ or $$ a = \frac{3 - \sqrt{5}}{2} $$
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