Question - Solving a Quadratic Equation by Factoring
mathsolving quadratic equationsquadratic formulafactoring quadratic expressionsquadratic equation factoring
Solution:
The equation in the image is a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. We will try to solve the equation by factoring first.The equation is:\[ 15x^2 - 11x - 38 = 0 \]For factoring, we need to find two numbers that multiply to $$ 15 \times (-38) = -570 $$ and add up to $$ -11 $$.The two numbers that fit these requirements are $$ -30 $$ and $$ 19 $$, since $$ -30 \times 19 = -570 $$ and $$ -30 + 19 = -11 $$.Now we can express $$ -11x $$ as $$ -30x + 19x $$:\[ 15x^2 - 30x + 19x - 38 = 0 \]Group the terms:\[ (15x^2 - 30x) + (19x - 38) = 0 \]Factor out common factors from each group:\[ 15x(x - 2) + 19(x - 2) = 0 \]Now factor out $$ (x - 2) $$ which is common:\[ (x - 2)(15x + 19) = 0 \]Finally, we set each factor equal to zero to find the solutions for $$ x $$:\[ x - 2 = 0 \quad \text{or} \quad 15x + 19 = 0 \]Solving each equation for $$ x $$ gives us:\[ x = 2 \quad \text{or} \quad x = -\frac{19}{15} \]So the solutions to the quadratic equation are $$ x = 2 $$ and $$ x = -\frac{19}{15} $$.
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