2x^2+2x>4
x^2 + 2x > 4
add 1 to both sides
x^2 + 2x + 1 > 5
(x+1)^2 > 5
x+1>√5 or -x-1 > √5
x > √5 - 1 OR -x > √5 + 1
x > √5-1 OR x < -√5-1
2x^2+2x>4 Divide both sides by 2
x ^ 2 + x > 2
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take one half of thje coefficient of x and square it.
in this case ( 1 / 2 ) ^ 2 = 1 / 4
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Add 1 / 4 to both sides
x ^ 2 + x + 1 / 4 > 2 + 1 / 4
x ^ 2 + x + 1 / 4 > 8 / 4 + 1 / 4
x ^ 2 + x + 1 / 4 > 9 / 4
Factor the left hand side
( x + 1 / 2 ) ^ 2 > 9 / 4
Take the square root of both sides
abs ( x + 1 / 2 ) > 3 / 2
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abs mean absolute value
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Solutions:
1 )
x + 1 / 2 > 3 / 2 Subtract 1 / 2 to both sides
x + 1 / 2 - 1 / 2 > 3 / 2 - 1 / 2
x > 2 / 2
x > 1
2 )
- x - 1 / 2 > 3 / 2 Add 1 / 2 to both sides
- x - 1 / 2 + 1 / 2 > 3 / 2 + 1 / 2
- x > 4 / 2
- x > 2 Multiply both sides by - 1
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If you multiply or divide the inequality by a negative number
you have to flip the inequality sign.
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In this case x < - 2
So solutions are:
x < - 2 and x > 1
sorry Kiki
don't know how I missed that 2 in front of 2x^2
Go with Bosnian's solution
or, how about this...
2x^2 + 2x > 4
x^2 + x - 2 > 0
(x+2)(x-1) > 0
(x+2>0 and x-1<0) OR ( x+2 < 0 and x-1 > 0 )
( x>-2 and x<1) OR x< -2 and x> 1
-2 < x < 1 OR null set
-2 < x < 1
To solve the inequality 2x^2 + 2x > 4, we need to find the values of x that make the inequality true. Here's how you can do it step by step:
1. Start by subtracting 4 from both sides of the inequality:
2x^2 + 2x - 4 > 0
2. Simplify the equation:
2x^2 + 2x - 4 - 4 > 0
2x^2 + 2x - 8 > 0
3. Now we have a quadratic inequality. To solve it, we need to factorize the quadratic equation:
2(x^2 + x - 4) > 0
4. Next, find the values of x that make each factor equal to zero. Set each factor to zero and solve for x:
x^2 + x - 4 = 0
This quadratic equation does not factorize easily, so we'll use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying the formula to our equation, we have:
x = (-1 ± √(1^2 - 4(1)(-4))) / (2(1))
x = (-1 ± √(1 + 16)) / 2
x = (-1 ± √17) / 2
So the two solutions for x are (-1 + √17) / 2 and (-1 - √17) / 2.
5. Now, we need to determine the intervals where the expression 2(x^2 + x - 4) is greater than zero. To do this, we'll consider the signs of the expression for different ranges of x.
a. Start by choosing a value of x less than the smaller root, such as x = -10:
2(-10^2 + (-10) - 4) > 0
2(100 - 10 - 4) > 0
2(86) > 0
172 > 0
Since the expression is positive for x = -10, it is positive for all x values less than the smaller root.
b. Next, consider a value of x between the two roots, such as x = 0:
2(0^2 + 0 - 4) > 0
2(-4) > 0
-8 > 0
Since the expression is negative for x = 0, it is negative for all x values between the two roots.
c. Finally, choose a value of x greater than the larger root, such as x = 10:
2(10^2 + 10 - 4) > 0
2(100 + 10 - 4) > 0
2(106) > 0
212 > 0
Since the expression is positive for x = 10, it is positive for all x values greater than the larger root.
6. Putting it all together, the solution to the inequality 2x^2 + 2x > 4 is:
x < (-1 - √17) / 2 or x > (-1 + √17) / 2
So, this means that the values of x that satisfy the inequality are all the values less than the smaller root (-1 - √17) / 2 or greater than the larger root (-1 + √17) / 2.