How do I solve this quadratic inequality? X2- 7x+10>0

the problem is actually written x to the 2 power -7x+10>0.

You factorize the quadratic funtion on the left.

x²-7x+10>0
(x-2)(x-5)>0

So the answer should be:

x<2 OR x>5

ײ+7×+10<0

ײ+7×+10=0
(×+2)(×+5)=0
×=-2 ×=-5
Test (0)
(×+ ) (×+ )

To solve the quadratic inequality x^2 - 7x + 10 > 0, you can use the following steps:

Step 1: Factorize the quadratic expression, if possible. However, in this case, the quadratic expression does not factorize nicely. Therefore, we need to use an alternative method.

Step 2: Find the roots of the equation x^2 - 7x + 10 = 0. To find the roots, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Using the coefficients from our equation, a = 1, b = -7, and c = 10, we can substitute them into the formula:

x = (7 ± √((-7)^2 - 4(1)(10))) / (2*1)

x = (7 ± √(49 - 40)) / 2

x = (7 ± √9) / 2

x = (7 ± 3) / 2

So, the roots of the equation are x = 5 and x = 2.

Step 3: Use the roots from Step 2 to create intervals on a number line.

On the number line, plot the root x = 5 on the right side and x = 2 on the left side.

Step 4: Test a point within each interval to determine the sign of the quadratic expression.

Let's test the point x = 0:

For x < 2: (0)^2 - 7(0) + 10 > 0, which gives 10 > 0, so the quadratic expression is positive in this interval.

For 2 < x < 5: (3)^2 - 7(3) + 10 > 0, which gives 1 > 0, so the quadratic expression is positive in this interval.

For x > 5: (6)^2 - 7(6) + 10 > 0, which gives -2 > 0, so the quadratic expression is negative in this interval.

Step 5: Write the solution in interval notation.

From the test results, we can conclude that the quadratic expression x^2 - 7x + 10 > 0 is positive in the intervals (-∞, 2) and (5, ∞).

Therefore, the solution in interval notation is (-∞, 2) U (5, ∞).

To solve the quadratic inequality x^2 - 7x + 10 > 0, you can follow these steps:

Step 1: Factorize the quadratic expression on the left side of the inequality, if possible. In this case, the quadratic expression cannot be easily factorized, so we need to use an alternative method.

Step 2: Find the x-intercepts of the quadratic equation x^2 - 7x + 10 = 0 by setting it equal to zero. This is done by factoring or by using the quadratic formula. The x-intercepts are the points where the quadratic equation intersects the x-axis.

Factoring:
(x - 2)(x - 5) = 0

Using the quadratic formula:
x = (-(-7) ± √((-7)^2 - 4(1)(10))) / (2(1))
x = (7 ± √(49 - 40)) / 2
x = (7 ± √9) / 2
x = (7 ± 3) / 2

So, the x-intercepts are x = 2 and x = 5.

Step 3: Graph the quadratic equation on a number line, plotting the x-intercepts as open circles and marking the regions between them.

2 5
o----------------o

Step 4: Choose a test point from each region and substitute it into the original inequality to determine the sign of the expression.

Let's choose x = 0 as a test point. Substituting into the inequality:
(0)^2 - 7(0) + 10 > 0
0 - 0 + 10 > 0
10 > 0

Since the inequality is true for the test point x = 0, the region to the left of 2 is part of the solution.

Next, let's choose x = 3 as a test point. Substituting into the inequality:
(3)^2 - 7(3) + 10 > 0
9 - 21 + 10 > 0
-2 > 0

Since the inequality is false for the test point x = 3, the region between 2 and 5 is not part of the solution.

Finally, let's choose x = 6 as a test point. Substituting into the inequality:
(6)^2 - 7(6) + 10 > 0
36 - 42 + 10 > 0
4 > 0

Since the inequality is true for the test point x = 6, the region to the right of 5 is part of the solution.

Step 5: Based on the tests, we can determine the solution to the quadratic inequality. The solution is the union of the regions where the inequality is true. In this case, the solution is x < 2 or x > 5.

Therefore, the solution to the quadratic inequality x^2 - 7x + 10 > 0 is x < 2 or x > 5.