Solve: x^2 +8x </=9.
How would I solve this?
First solve x^2 +8x -9 = 0
(x+9)(x-1) = 0
The solutions are x = 1 and -9
In the region -9 < x < 1, x^2 + 8x <9
Thanks just to make sure I have the problem as x^2+8x less than or equal to 9.
In that case -9 </= x </= 1 is the solution
To solve the inequality x^2 + 8x ≤ 9, you can follow these steps:
Step 1: Begin by rearranging the inequality to get a quadratic expression in standard form, which is in the form of ax^2 + bx + c ≤ 0.
x^2 + 8x - 9 ≤ 0
Step 2: Next, factor the quadratic expression if possible. However, in this case, the expression cannot be factored easily, so we'll use an alternative method.
Step 3: Set the quadratic expression equal to zero and find the x-intercepts (also known as the roots) by solving the equation x^2 + 8x - 9 = 0. You can use factoring, completing the square, or the quadratic formula.
In this case, the quadratic equation does not factor easily, so you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our quadratic equation x^2 + 8x - 9 = 0, a = 1, b = 8, and c = -9. Plug these values into the formula:
x = (-8 ± √(8^2 - 4(1)(-9))) / (2(1))
x = (-8 ± √(64 + 36)) / 2
x = (-8 ± √100) / 2
x = (-8 ± 10) / 2
This gives us two possible solutions:
x1 = (-8 + 10) / 2 = 1
x2 = (-8 - 10) / 2 = -9
So the roots of the quadratic equation are x = 1 and x = -9.
Step 4: Plot these roots on a number line and check the intervals between the roots to determine when the quadratic expression is less than or equal to zero.
On a number line, mark the points x = -9 and x = 1. These points divide the number line into three intervals: (-∞, -9), (-9, 1), and (1, ∞).
Step 5: Finally, choose a test point in each of the intervals and substitute it into the quadratic expression. Determine whether the expression is less than or equal to zero or greater than zero.
For example, let's choose -10, 0, and 2 as test points.
For x = -10: (-10)^2 + 8(-10) - 9 = 100 - 80 - 9 = 11. Since 11 is positive, the expression is greater than zero in the interval (-∞, -9).
For x = 0: (0)^2 + 8(0) - 9 = -9. Since -9 is negative, the expression is less than zero in the interval (-9, 1).
For x = 2: (2)^2 + 8(2) - 9 = 4 + 16 - 9 = 11. Since 11 is positive, the expression is greater than zero in the interval (1, ∞).
Step 6: Based on the sign of the expression, we conclude that the inequality x^2 + 8x ≤ 9 is true when x is in the interval (-9, 1], which means x is greater than or equal to -9 and less than or equal to 1.