Worked Solution & Example Answer:Solve $2x^2 + 3x - 2 > 0$ - Edexcel - GCSE Maths - Question 19 - 2017 - Paper 3
Step 1
Factor the quadratic equation
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Answer
To solve the inequality 2x2+3x−2>0, we first factor the quadratic expression. We can check for factors of the form: ( (2x + 1)(x - 2) ). Thus, we can express the inequality as:
(2x+1)(x−2)>0
Step 2
Determine the critical points
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Answer
Setting each factor to zero gives us the critical points:
2x+1=0⇒x=−21
x−2=0⇒x=2
The critical points are x=−21 and x=2.
Step 3
Test intervals and find solution
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Answer
We now test the intervals determined by the critical points: ( (-\infty, -\frac{1}{2}), (-\frac{1}{2}, 2), (2, \infty) ).
For ( x < -\frac{1}{2} ) (e.g., ( x = -1 )):
(2(−1)+1)((−1)−2)=(−2+1)(−3)=1>0 (satisfies the inequality)
For ( -\frac{1}{2} < x < 2 ) (e.g., ( x = 0 )):
(2(0)+1)(0−2)=(1)(−2)=−2<0 (does not satisfy the inequality)
For ( x > 2 ) (e.g., ( x = 3 )):
(2(3)+1)(3−2)=(6+1)(1)=7>0 (satisfies the inequality)
Thus, we combine the results and conclude that the solution for the inequality is:
