To solve this quadratic equation, we can either factor the equation or use the quadratic formula.
Factoring:
To factor the quadratic equation 5x^2 - 7x - 2 = 0, we need to find two numbers that multiply to -10 (product of the coefficient of x^2 and the constant term) and add up to -7 (coefficient of x). The two numbers are -10 and 1.
So, we rewrite the middle term as -10x + 3x:
5x^2 - 10x + 3x - 2 = 0
5x(x - 2) + 1(3x - 2) = 0
(5x + 1)(x - 2) = 0
Setting each factor to zero:
5x + 1 = 0
5x = -1
x = -1/5
x - 2 = 0
x = 2
Therefore, the solutions are x = -1/5 and x = 2.
Using the Quadratic Formula:
Alternatively, we can use the quadratic formula to solve for x:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 5, b = -7, and c = -2.
Plugging in the values:
x = (7 ± √((-7)^2 - 4(5)(-2))) / 2(5)
x = (7 ± √(49 + 40)) / 10
x = (7 ± √89) / 10
Therefore, the solutions are x = (7 + √89) / 10 and x = (7 - √89) / 10.
5x^2-7x-2=0
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