Question - Solving a Quadratic Equation by Factoring
mathsolving quadratic equationquadratic formulafactoring quadratic equationsolving equations by factoring
Solution:
The equation provided in the image is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. Since the coefficients are quite large, I'll solve it by factoring for simplicity.The equation is:\[ 28x^2 - 72x - 180 = 0 \]To make it easier to factor, we can look for a common factor between the coefficients, which in this case is 4. Let's divide the whole equation by 4 to simplify:\[ 7x^2 - 18x - 45 = 0 \]Now, we look to factor the quadratic. We need to find two numbers that multiply to $$7 \times (-45)$$ (which is -315) and at the same time add up to -18 (the coefficient of the x term).These numbers are -25 and +7. So, we can express the middle term -18x as -25x + 7x, like this:\[ 7x^2 - 25x + 7x - 45 = 0 \]Group the terms to factor by grouping:\[ (7x^2 - 25x) + (7x - 45) = 0 \]Factor out the common factors:\[ x(7x - 25) + 5(7x - 25) = 0 \]Now you can see that $$ (7x - 25) $$ is a common factor:\[ (x + 5)(7x - 25) = 0 \]To find the solutions for x, set each factor equal to zero:\[ x + 5 = 0 \quad \text{or} \quad 7x - 25 = 0 \]Solve both equations:\[ x = -5 \quad \text{or} \quad 7x = 25 \]\[ \quad \text{or} \quad x = \frac{25}{7} \approx 3.571 \]So, the solutions to the quadratic equation are:\[ x = -5 \quad \text{or} \quad x = \frac{25}{7} \]
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