Answer:
Explanation:
To find the roots of the equation x^2-8x-20=0 using the square roots method, we can follow these steps:
Step 1: Identify the coefficients of the quadratic equation. In this case, the coefficient of x^2 is 1, the coefficient of x is -8, and the constant term is -20.
Step 2: Use the quadratic formula to find the roots of the equation. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation.
In our equation, a = 1, b = -8, and c = -20. Plugging these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(1)(-20))) / (2(1))
Simplifying further, we have:
x = (8 ± √(64 + 80)) / 2
x = (8 ± √144) / 2
Step 3: Simplify the square root. √144 = 12, so we have:
x = (8 ± 12) / 2
Step 4: Split the equation into two separate equations, one with the positive square root and one with the negative square root:
x = (8 + 12) / 2 and x = (8 - 12) / 2
Simplifying further, we get:
x = 20 / 2 and x = -4 / 2
x = 10 and x = -2
Therefore, the roots of the equation x^2-8x-20=0 are x = 10 and x = -2.
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Now, to reinforce your understanding, let's practice solving a similar problem:
Example problem: Find the roots of the equation 2x^2 + 5x - 3 = 0 using the square roots method.
Step 1: Identify the coefficients of the quadratic equation. In this case, the coefficient of x^2 is 2, the coefficient of x is 5, and the constant term is -3.
Step 2: Use the quadratic formula to find the roots of the equation. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation.
In our equation, a = 2, b = 5, and c = -3. Plugging these values into the quadratic formula, we get:
x = (-(5) ± √((5)^2 - 4(2)(-3))) / (2(2))
Simplifying further, we have:
x = (-5 ± √(25 + 24)) / 4
x = (-5 ± √49) / 4
Step 3: Simplify the square root. √49 = 7, so we have:
x = (-5 ± 7) / 4
Step 4: Split the equation into two separate equations, one with the positive square root and one with the negative square root:
x = (-5 + 7) / 4 and x = (-5 - 7) / 4
Simplifying further, we get:
x = 2 / 4 and x = -12 / 4
x = 1/2 and x = -3
Therefore, the roots of the equation 2x^2 + 5x - 3 = 0 are x = 1/2 and x = -3.
Now, it's your turn to practice. Try solving the following problem:
Practice problem: Find the roots of the equation 3x^2 - 6x + 3 = 0 using the square roots method.
Take your time to solve the problem, and once you're ready, you can ask for guidance or submit your answer for grading.