A quadratic equation is an equation that can be written in the standard form

ax+ bx + c = 0, where a, b, and c are real numbers and a¹ 0.

Solving quadratic equations by factoring

When a quadratic equation is in standard form and the nonzero side, ax+ bx + c , can be factored, the equation can be solved by factoring. To solve by factoring, the following property is used:

Zero-Factor Property: If a and b are complex numbers, with a × b = 0,

then a = 0 or b = 0 or both equal zero.

Example. Solve: 2x + 7x = 4

Solution.

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 = k can be solved using the following property:

Square Root Property: The solution set of x= k is

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, the equation must be written in the form (x + n) = k. The following steps are used to solve the equation

ax+ bx + c = 0 , a ¹ 0 , by completing the square:

1. If a ¹ 1, divide both sides of the equation by a.

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

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

equation.

4. Factor the resulting trinomial as a perfect square; combine like 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 equation using 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 coefficient of x, b is the coefficient of x,

and c is the constant.

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

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

The solution set is