Now, we can rewrite the inequality in standard form:
-a^2 + 4a - 2 - 5 > 0 -a^2 + 4a - 7 > 0
Next, we can factor the quadratic equation:
-(a^2 - 4a + 7) > 0
Since the quadratic equation does not factor further, we can find the solutions by using the formula for the discriminant (b^2 - 4ac) to determine whether the quadratic equation has real roots. The discriminant is:
b^2 - 4ac = (-4)^2 - 4(-1)(7) = 16 + 28 = 44
Since the discriminant is positive, the quadratic equation has real roots. Therefore, we can solve the inequality by finding the roots of the quadratic equation and determining the sign of the inequality in each interval.
To find the roots, we can use the quadratic formula:
a = 1, b = -4, c = -7
Using the quadratic formula:
a = (-(-4) +/- sqrt((-4)^2 - 4(1)(-7))) / 2(1) a = (4 +/- sqrt(16 + 28)) / 2 a = (4 +/- sqrt(44)) / 2 a = (4 +/- 2sqrt(11)) / 2 a = 2 +/- sqrt(11)
Therefore, the roots of the quadratic equation are 2 + sqrt(11) and 2 - sqrt(11).
Now, we can determine the sign of the inequality in each interval by testing a value in each interval:
Test a value less than 2 - sqrt(11): a = 0 -0 - 4(0) + 7 = 7, which is positive Therefore, the inequality holds in this interval.
Test a value between 2 - sqrt(11) and 2 + sqrt(11): a = 2 -4(2) + 4(2) - 7 = -7, which is negative Therefore, the inequality does not hold in this interval.
Test a value greater than 2 + sqrt(11): a = 4 -16 + 4(4) - 7 = 1, which is positive Therefore, the inequality holds in this interval.
Therefore, the solution to the inequality -a^2 + 4a - 7 > 0 is a < 2 - sqrt(11) or a > 2 + sqrt(11).
To solve the inequality (a-2)-a^2 > 5-3a, we can first simplify the left side of the inequality:
(a-2) - a^2 > 5 - 3a
a - 2 - a^2 > 5 - 3a
a - a^2 - 2 > 5 - 3a
-a^2 + a - 2 > 5 - 3a
-a^2 + 4a - 2 > 5
Now, we can rewrite the inequality in standard form:
-a^2 + 4a - 2 - 5 > 0
-a^2 + 4a - 7 > 0
Next, we can factor the quadratic equation:
-(a^2 - 4a + 7) > 0
Since the quadratic equation does not factor further, we can find the solutions by using the formula for the discriminant (b^2 - 4ac) to determine whether the quadratic equation has real roots. The discriminant is:
b^2 - 4ac = (-4)^2 - 4(-1)(7) = 16 + 28 = 44
Since the discriminant is positive, the quadratic equation has real roots. Therefore, we can solve the inequality by finding the roots of the quadratic equation and determining the sign of the inequality in each interval.
To find the roots, we can use the quadratic formula:
a = 1, b = -4, c = -7
Using the quadratic formula:
a = (-(-4) +/- sqrt((-4)^2 - 4(1)(-7))) / 2(1)
a = (4 +/- sqrt(16 + 28)) / 2
a = (4 +/- sqrt(44)) / 2
a = (4 +/- 2sqrt(11)) / 2
a = 2 +/- sqrt(11)
Therefore, the roots of the quadratic equation are 2 + sqrt(11) and 2 - sqrt(11).
Now, we can determine the sign of the inequality in each interval by testing a value in each interval:
Test a value less than 2 - sqrt(11): a = 0
-0 - 4(0) + 7 = 7, which is positive
Therefore, the inequality holds in this interval.
Test a value between 2 - sqrt(11) and 2 + sqrt(11): a = 2
-4(2) + 4(2) - 7 = -7, which is negative
Therefore, the inequality does not hold in this interval.
Test a value greater than 2 + sqrt(11): a = 4
-16 + 4(4) - 7 = 1, which is positive
Therefore, the inequality holds in this interval.
Therefore, the solution to the inequality -a^2 + 4a - 7 > 0 is a < 2 - sqrt(11) or a > 2 + sqrt(11).