Question - Solving Quadratic Equations by Factoring
Solution:
The equation provided in the image is a quadratic equation of the form $$ ax^2 + bx + c = 0 $$, where $$ a = 6 $$, $$ b = -7 $$, and $$ c = -10 $$. To solve this equation, we can either factor the quadratic, complete the square, or use the quadratic formula.Let's attempt factoring first. We need two numbers that multiply to $$ a \times c = 6 \times (-10) = -60 $$ and that add to $$ b = -7 $$.Looking for such numbers, we find that -12 and +5 multiply to -60 and add to -7.Now we can write:$$6x^2 - 12x + 5x - 10 = 0$$Group the terms to factor by grouping:$$ (6x^2 - 12x) + (5x - 10) = 0 $$Factor out common factors:$$ 6x(x - 2) + 5(x - 2) = 0 $$Now we can see that $$ (x - 2) $$ is a common factor:$$ (6x + 5)(x - 2) = 0 $$Using the zero product property, we can set each factor equal to zero:$$ 6x + 5 = 0 $$ or $$ x - 2 = 0 $$Solving for $$ x $$ gives us:$$ x = -\frac{5}{6} $$ or $$ x = 2 $$So the solutions to the quadratic equation are $$ x = -\frac{5}{6} $$ and $$ x = 2 $$.
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