x Substituted Evaluate True Statement?
−3/4
−1/2
1/2
For the quadratic equation −2x^2−3x+2=0
, complete the table by testing each of the given values to determine whether it is a solution. Identify which one of the values is in the solution set.
To determine if a given value is a solution to the quadratic equation, we substitute it into the equation in place of x and simplify. If we obtain a true statement (the left-hand side equals the right-hand side), then the substituted value is a solution to the equation.
The equation we are testing is:
−2x^2 − 3x + 2 = 0
Let's substitute the given values one by one:
1. Substituting x = −3/4:
−2(−3/4)^2 − 3(−3/4) + 2
= −2(9/16) + 9/4 + 2
= −(9/8) + 9/4 + 2
= −(9/8) + (18/8) + (16/8)
= (18/8 − 9/8 + 16/8)
= (25/8)
Since 25/8 does not equal 0, the value x = −3/4 is not a solution to the equation.
2. Substituting x = −1/2:
−2(−1/2)^2 − 3(−1/2) + 2
= −2(1/4) + 3/2 + 2
= −(1/2) + 3/2 + 2
= 1 + 2
= 3
Since 3 does not equal 0, the value x = −1/2 is not a solution to the equation.
3. Substituting x = 1/2:
−2(1/2)^2 − 3(1/2) + 2
= −2(1/4) − 3/2 + 2
= −(1/2) − 3/2 + 2
= −4/2 + 2
= −2 + 2
= 0
Since we have obtained 0 after the substitution, the statement is true, and thus x = 1/2 is a solution to the equation −2x^2 − 3x + 2 = 0.
Summarizing our findings in the table:
x | Substituted | Evaluate | True Statement?
-----------------------------------------------------
−3/4 | −2(−3/4)^2... | 25/8 | False
−1/2 | −2(−1/2)^2... | 3 | False
1/2 | −2(1/2)^2... | 0 | True
The value in the solution set is x = 1/2.