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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  