Solving a Quadratic Equation by Factoring
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 \( x = -\frac{19}{15} \).
