Question

Verify that \(x=-2\) and \(x=-3\) are both solutions of \(x^{2}+5 x+6=0\)

Short answer

Answer: Yes, both -2 and -3 are solutions to the given quadratic equation.

Step-by-step solution

Substitute x = -2 into the equation

Substitute x = -2 into the given equation: \((-2)^{2} + 5(-2) + 6\)

Simplify and solve for x = -2

Solve the equation: \((4) + (-10) + 6 = 0\) \(-6 + 6 = 0\) \(0 = 0\) The equation holds true when x = -2.

Substitute x = -3 into the equation

Substitute x = -3 into the given equation: \((-3)^{2} + 5(-3) + 6\)

Simplify and solve for x = -3

Solve the equation: \((9) + (-15) + 6 = 0\) \(-6 + 6 = 0\) \(0 = 0\) The equation holds true when x = -3. As both x = -2 and x = -3 satisfy the equation \(x^{2}+5x+6=0\), we can verify that \(x=-2\) and \(x=-3\) are both solutions of the given quadratic equation.

Key concepts

Solving Quadratic Equations

When we talk about solving quadratic equations, we're referring to finding values for the variable that make the equation true. Quadratic equations always take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The solutions to the quadratic equation are often called the "roots" of the equation. In simpler terms, solving a quadratic equation means finding the values of \( x \) that "work" when plugged back into the equation to make both sides equal.

There are several methods to solve quadratic equations:

• Factoring: involves writing the equation as a product of two binomials.
• Quadratic Formula: solves any quadratic equation using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the Square: rearranges the equation to create a perfect square trinomial.
• Graphing: plotting the quadratic function and identifying where it crosses the x-axis.

Each method requires different approaches, but they all aim to find the same result: the roots of the equation.

Polynomial Roots

The roots of a polynomial are the solutions to the equation formed by setting the polynomial equal to zero. In the context of quadratic equations, these roots represent the specific \( x \) values at which the equation equals zero. For a standard quadratic equation \( ax^2 + bx + c = 0 \), there are usually two roots.

These roots can be real or complex numbers depending on the value of the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the roots are complex and occur as a conjugate pair.

In our given example, \( x^2 + 5x + 6 = 0 \), we found the roots to be \( x = -2 \) and \( x = -3 \). Both are real and distinct because the discriminant \( 25 - 24 = 1 \) is greater than zero.

Verification of Solutions

Verification of solutions is a critical step in solving equations, ensuring the solutions are correct. This process involves substituting the found solutions back into the original equation and checking to see if the equation is satisfied.

For our exercise, we substituted \( x = -2 \) and \( x = -3 \) back into the equation \( x^2 + 5x + 6 = 0 \). In both cases, when the respective \( x \) value was inserted, the left-hand side simplified to zero, confirming that each solution was valid:

• For \( x = -2 \):
  \((-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0\)
• For \( x = -3 \):
  \((-3)^2 + 5(-3) + 6 = 9 - 15 + 6 = 0\)

Each step of our calculations returned zero, affirming our solutions are indeed correct. This process of checking guarantees the integrity and correctness of the result, highlighting the importance of accuracy in mathematical operations.