If the roots of the equation x² + 7x + p = 0 are -3 and -4, find p. (2019)

If the roots of the equation x² + 7x + p = 0 are -3 and -4, find p. (2019)

Step 1: Identify the given quadratic equation, which is x² + 7x + p = 0.
Step 2: Note the roots of the equation, which are -3 and -4.
Step 3: Calculate the sum of the roots: -3 + -4 = -7.
Step 4: The sum of the roots can also be expressed as -b/a, where b is the coefficient of x. Here, b = 7 and a = 1, so -b/a = -7.
Step 5: Since the sum of the roots matches, we can proceed to find the product of the roots.
Step 6: Calculate the product of the roots: -3 * -4 = 12.
Step 7: The product of the roots can also be expressed as c/a, where c is the constant term (p in this case) and a is the coefficient of x². Here, c = p and a = 1, so c/a = p.
Step 8: Since the product of the roots is 12, we have p = 12.

Quadratic Equations – Understanding the relationship between the coefficients of a quadratic equation and its roots through Vieta's formulas.
Sum and Product of Roots – Applying the properties of the sum and product of the roots of a quadratic equation to find unknown coefficients.