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2x + 7x = 4 | |

2x + 7x – 4 = 0 | To get the equation in standard form, subtract 4 from each side; one side must be 0 when solving by factoring |

(2x – 1)(x + 4) = 0 | Factor left side of equation |

2x – 1 = 0 or x + 4 = 0 | Set each factor equal to 0, using the Zero Factor Property |

2x = 1 x = -4 x = | Solve each of the resulting two equations |

The solution set is

**Solving Quadratic Equations Using the Square Root Property**

A quadratic equation of the form x = kcan be solved using the following property:

**Square Root Property**: The solution set of x= kis

Example. Solve: a) 4x = 20

b) z = -49

Solution. a)

4x = 20 | |

Divide both sides by 4 to isolate x | |

Simplify | |

x = ± | Use the square root property to obtain two solutions |

The solution set is

b)

z= -49 | |

z = | Use the square root property |

z = ± 7i | Simplify the radical |

The solution set is

**Solving Quadratic Equations by Completing the Square**

To solve a quadratic equation by completing the square, theequation must be written in the form (x + n) = k.The following steps are used to solve the equation

ax+ bx + c = 0 , a ¹0 , by completing the square:

1. If a ¹ 1, divide both sidesof the equation by a.

2. Rewrite the equation so that the constant term is isolatedon one side of the equation.

3. Take half the coefficient of x and square this result; addthis square to both sides of the

equation.

4. Factor the resulting trinomial as a perfect square; combinelike terms on the other

side.

5. Use the square root property to complete the solution.

Example. Solve by completing the square: 2x + 5x– 4 = 0.

Solution.

2x + 5x – 4 = 0 | |

Divide both sides by 2 | |

Add 2 to both sides to isolate the constant |

Take half the coefficient of x and square it:

Add to both sides of the equation | |

Factor the trinomial; add the constants | |

Use the square root property | |

Add - to both sides and simplify the radical |

The solution set is

**Solving Quadratic Equations Using the Quadratic Formula**

The solutions of the quadratic equation ax+ bx+ c = 0 , a ¹ 0, are

The following steps are used to solve a quadratic equationusing the quadratic formula:

1. Write the equation in standard form, ax+ bx+ c = 0.

2. Determine the values of a, b, and c; a is the coefficientof x, b is the coefficient of x,

and c is the constant.

3. Substitute these values of a, b, and c into the quadraticformula.

4. Simplify.

Example. Solve using the quadratic formula: 3x = 2x– 4

Solution.

3x = 2x – 4 | |

3x - 2x + 4 = 0 | Write the equation in standard form. |

a = 3, b = -2, c = 4 | Determine the values of a, b, and c |

The quadratic formula | |

Substitute the values of a, b, and c into the formula | |

Simplify | |