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.
- 2x + 4 = 0
