Math Problem Statement

Factorise and then solve the quadratic equation z^2 + 10z + 25 = 0.

Solution

The quadratic equation provided is:

z2+10z+25=0z^2 + 10z + 25 = 0

Step 1: Factorize the quadratic equation

This is a perfect square trinomial. To factorize it:

z2+10z+25=(z+5)(z+5)=(z+5)2z^2 + 10z + 25 = (z + 5)(z + 5) = (z + 5)^2

Step 2: Solve the equation

Since (z+5)2=0(z + 5)^2 = 0, we can solve by setting the factor equal to zero:

z+5=0z + 5 = 0

z=5z = -5

Thus, the solution to the quadratic equation is z=5z = -5, which is a repeated solution.

Final Answer:

  • z=5z = -5
  • z=5z = -5 (since it's a repeated solution)

Would you like more details or have any other questions?


Here are 5 related questions to deepen understanding:

  1. What is the general method to solve any quadratic equation?
  2. How do you recognize a perfect square trinomial?
  3. What is the significance of repeated solutions in a quadratic equation?
  4. How can you use the quadratic formula to solve this same equation?
  5. What are the characteristics of the graph of a quadratic equation with a repeated solution?

Tip: When you factor a quadratic equation, always check if it's a perfect square trinomial to simplify the process!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Factorization

Formulas

Quadratic equation formula: ax^2 + bx + c = 0

Theorems

Perfect Square Trinomial
Zero-Product Property

Suitable Grade Level

Grades 7-9