SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE
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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