Solve the Quadratic Inequality: x²+7x+10<0 Step-by-Step

To solve the inequality $x^2 + 7x + 10 < 0$, follow these steps:

• Step 1: Factor the quadratic expression.
• Step 2: Identify the roots of the quadratic equation.
• Step 3: Determine the sign of the expression in the intervals defined by the roots.

Step 1: Factor the quadratic expression:

$x^2 + 7x + 10$ can be factored into $(x + 5)(x + 2)$.
This is because $(x + 5)(x + 2) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10$.

Step 2: Identify the roots:

The roots are found by setting the factored expression to zero:
$x + 5 = 0$ or $x + 2 = 0$, which gives $x = -5$ and $x = -2$.

Step 3: Determine the sign of the expression in the intervals:

The critical points divide the number line into three intervals: $x < -5$, $-5 < x < -2$, and $x > -2$.

Test a point in each interval to determine where the product is negative:

• For $x < -5$, choose $x = -6$: $(x + 5)(x + 2) = (-6 + 5)(-6 + 2) = (-1)(-4) = 4$, which is positive.
• For $-5 < x < -2$, choose $x = -3$: $(x + 5)(x + 2) = (-3 + 5)(-3 + 2) = (2)(-1) = -2$, which is negative.
• For $x > -2$, choose $x = 0$: $(x + 5)(x + 2) = (0 + 5)(0 + 2) = 10$, which is positive.

The expression is negative only in the interval $-5 < x < -2$.

Therefore, the solution to the inequality is $-5 < x < -2$.