Category Archives: 1.4 Quadratic Equations

Example: Solve the quadratic equation with the quadratic formula.

2x^2+7=4x

Solution:

The original equation.
Write the equation so that all of the terms are on the same side.
Identify a, b and c.a=2, b=-4, c=7
The quadratic formula.	$x={-b pm sqrt{b^2-4ac}}/{2a}$
Substitute the values into the quadratic formula.	$x={-b pm sqrt{b^2-4ac}}/{2a}$ $x={-(-4) pm sqrt{(-4)^2-4(2)(7)}}/{2(2)}$
Simplify using order of operations by applying powers, multiplying and then subtracting.	$x={-(-4) pm sqrt{(-4)^2-4(2)(7)}}/{2(2)}$
Simplify the radical by looking for perfect square factors of 40.
Simplify by canceling the common factor of 2 out of the terms in the numerator and denominator.

The solutions to the quadratic equation are x={2 + i sqrt{10}}/{2} and x={2 - i sqrt{10}}/{2} .

Solving Quadratic Equations by Factoring: Trinomial a=1

Example: Solve the quadratic equation by factoring.

x^2+x=20

Solution:

The original equation
Write the equation with all the terms on one side of the equation and zero on the other side of the equation.
Factor the expression on one side.
Use the zero product property and set each factor equal to zero.	or
Solve each equation.	or or or

Check: x=4

x^2+x=20
(4)^2+4=20
16+4=20
20=20

Since the value of 4 makes the equation true, 4 is a solution to the equation.

Check: x=-5

x^2+x=20
(-5)^2+(-5)=20
25-5=20
20=20

Since the value of -5 makes the equation true, -5 is a solution to the equation.

Example: Solve the quadratic equation by factoring.

Original Equation
The equation already has all of the terms on the same side and zero on the other side.
Factor the expression on one side. The expression for this equation is a difference of squares.
Use the zero product property and set each factor equal to zero.	or
Solve each equation.	or or or

Example: Solve the quadratic equation with completing the square.

x^2-9=0

The original quadratic equation.
Rewrite the quadratic equation so that the square and everything that the square applies to are on one side of the equation. This is called isolating the square.
Cancel out the square by square rooting both sides.	$sqrt{x^2}=sqrt{9}$
The remaining equations are already solved. The solutions to the equation are 3 and -3.	or

Example: Solve the quadratic equation by completing the square

2x^2+8x-9=0

Solution:

Original quadratic equation
Move the constant to the other side of the equation by adding 9 to both sides.
Make the coefficient of the squared term 1 by dividing both sides of the equation by 2.	${2x^2+8x}/2=9/2$ ${2x^2}/2+{8x}/2=9/2$
Add the number that completes the square to both sides of the equation. The number that completes the square can be found by starting with the coefficient of the x term, dividing by 2 and then squaring.
The left hand side factors to be a perfect square
Simplify the right hand side by finding a common denominator and adding the fractions.	$(x+2)^2=9/2+{42}/{12}$
Solve the equation using the square root method. Start by taking the square root of both sides to get rid of the square	$sqrt{(x+2)^2}=sqrt{17/2}$
Solve for x by subtracting 2 from both sides.
Rationalize the denominator by applying the radical to the numerator and denominator
Rationalize the denominator by multiplying by the fraction