Algebra Chapter 10: Factoring Quadratic Equations

Factoring

• Factorization is a method of solving quadratic equations by expressing them as a product of binomials.
• A quadratic equation can be factored if it can be written in the form: ax^2 + bx + c = (x + m)(x + n)
• To factor a quadratic equation:
  • Look for two numbers whose product is ac and whose sum is b.
  • Rewrite the equation in the form ax^2 + bx + c = (x + m)(x + n)
• Example: x^2 + 5x + 6 = (x + 3)(x + 2)

Solving Quadratic Equations

• Quadratic equations can be solved using various methods:
  • Factoring: When the equation can be expressed as a product of binomials.
  • Quadratic Formula: A general method for solving quadratic equations.
  • Graphing: By plotting the graph of the related function and finding the x-intercepts.
• A quadratic equation can have:
  • Two distinct real roots
  • One repeated real root
  • Two complex conjugate roots

Quadratic Formula

• The Quadratic Formula is a general method for solving quadratic equations of the form ax^2 + bx + c = 0.
• The Quadratic Formula is: x = (-b ± √(b^2 - 4ac)) / 2a
• To use the Quadratic Formula:
  • Identify the values of a, b, and c from the equation.
  • Plug these values into the formula.
  • Simplify the expression to find the roots.
• The Quadratic Formula can be used to find the roots of any quadratic equation, even if it cannot be factored.

Factoring Quadratic Equations

• Factorization is a method of solving quadratic equations by expressing them as a product of binomials.
• A quadratic equation can be factored if it can be written in the form: ax^2 + bx + c = (x + m)(x + n)
• To factor a quadratic equation, look for two numbers whose product is ac and whose sum is b.
• Then, rewrite the equation in the form ax^2 + bx + c = (x + m)(x + n)
• Example: x^2 + 5x + 6 = (x + 3)(x + 2)

Solving Quadratic Equations

• Quadratic equations can be solved using various methods, including factoring, quadratic formula, and graphing.
• A quadratic equation can have two distinct real roots, one repeated real root, or two complex conjugate roots.

Quadratic Formula

• The Quadratic Formula is a general method for solving quadratic equations of the form ax^2 + bx + c = 0.
• The Quadratic Formula is: x = (-b ± √(b^2 - 4ac)) / 2a
• To use the Quadratic Formula, identify the values of a, b, and c from the equation, plug them into the formula, and simplify the expression to find the roots.
• The Quadratic Formula can be used to find the roots of any quadratic equation, even if it cannot be factored.

Factoring Quadratic Equations

• Factoring is a method for solving quadratic equations of the form ax^2 + bx + c = 0
• The goal of factoring is to express the quadratic expression as a product of two binomials
• Factoring is only possible when the equation has two rational roots
• To factor a quadratic equation, find two numbers whose product is ac and whose sum is b, then rewrite the middle term as the sum of these two numbers and factor by grouping

Quadratic Formula

• The quadratic formula is a general method for solving quadratic equations of the form ax^2 + bx + c = 0
• The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a
• The quadratic formula always gives two solutions for the value of x
• The quadratic formula is useful when the equation cannot be factored

Solving Quadratic Equations

• Quadratic equations can be solved using factoring, the quadratic formula, or graphing
• Graphing involves plotting the related function on a coordinate plane and finding the x-intercepts
• Solving quadratic equations can be used to model real-world problems, such as projectile motion, optimization problems, and electrical circuits
• Quadratic equations can have two distinct real roots, one repeated real root, or two complex conjugate roots