Solve $x^2+15 x+36=0$
$x=\{ \square \}$
Answer (1)
Factor the quadratic expression: Find two numbers that multiply to 36 and add up to 15, which are 3 and 12.
Rewrite the equation: ( x + 3 ) ( x + 12 ) = 0 .
Set each factor to zero: x + 3 = 0 or x + 12 = 0 .
Solve for x : The solutions are x = − 12 and x = − 3 , so x = { − 12 , − 3 } .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 15 x + 36 = 0 . Our goal is to find the values of x that satisfy this equation. We can solve this by factoring the quadratic expression.
Factoring the Quadratic To factor the quadratic expression x 2 + 15 x + 36 , we need to find two numbers that multiply to 36 and add up to 15. These numbers are 3 and 12, since 3 × 12 = 36 and 3 + 12 = 15 .
Rewriting the Equation Now we can rewrite the quadratic equation as ( x + 3 ) ( x + 12 ) = 0 .
Setting Factors to Zero To find the solutions for x , we set each factor equal to zero: x + 3 = 0 or x + 12 = 0 .
Solving for x Solving for x in each case, we get x = − 3 or x = − 12 .
Final Answer Therefore, the solutions to the quadratic equation x 2 + 15 x + 36 = 0 are x = − 3 and x = − 12 . So, x = { − 12 , − 3 } .
Examples
Quadratic equations are used in many real-world applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, and modeling various physical phenomena. For example, if you throw a ball, the height of the ball over time can be modeled by a quadratic equation. Solving the equation helps determine when the ball will hit the ground.
