Category Archives: 1.4 Quadratic Equations

Solving a Quadratic Equation using the Quadratic Formula: Example 1 of 1

Example:  Solve the quadratic equation with the quadratic formula.

2x^2+7=4x

Solution:

The original equation.
2x^2+7=4x
Write the equation so that all of the terms are on the same side.
2x^2+7=4x
2x^2+7-4x=4x-4x
2x^2-4x+7=0
Identify a, b and c.
a=2, b=-4, c=7
2x^2-4x+7=0
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)}
x={4 pm sqrt{16-56}}/{4}
x={4 pm sqrt{-40}}/{4}
Simplify the radical by looking for perfect square factors of 40.
x={4 pm sqrt{-40}}/{4}
x={4 pm sqrt{-1(4)(10)}}/{4}
x={4 pm 2i sqrt{10}}/{4}
Simplify by canceling the common factor of 2 out of the terms in the numerator and denominator.
x={4 pm 2i sqrt{10}}/{4}
x={2(2) pm 2i sqrt{10}}/{2(2)}
x={2 pm i sqrt{10}}/{2}

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
x^2+x=20
Write the equation with all the terms on one side of the equation and zero on the other side of the equation. x^2+x=20
x^2+x-20=20-20
x^2+x-20=0
Factor the expression on one side. x^2+x-20=0
(x+5)(x-4)=0
Use the zero product property and set each factor equal to zero. (x+5)(x-4)=0
x+5=0 or x-4=0
Solve each equation. x+5=0 or x-4=0
x+5-5=0-5 or x-4+4=0+4
x=-5 or x=4

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.

Solving a Quadratic Equation: The Square Root Method Example 1 of 1

Example: Solve the quadratic equation with completing the square.

x^2-9=0

The original quadratic equation.
x^2-9=0
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. x^2-9=0
x^2-9+9=0+9
x^2=9
Cancel out the square by square rooting both sides. x^2=9
sqrt{x^2}=sqrt{9}
x=pm 3
The remaining equations are already solved.  The solutions to the equation are 3 and -3. x=-3 or x=3

Quadratic Equation: Completing the Square

Example: Solve the quadratic equation by completing the square

2x^2+8x-9=0

Solution:

Original quadratic equation
2x^2+8x-9=0
Move the constant to the other side of the equation by adding 9 to both sides.
2x^2+8x-9=0
2x^2+8x-9+9=0+9
2x^2+8x=9
Make the coefficient of the squared term 1 by dividing both sides of the equation by 2.
2x^2+8x=9
{2x^2+8x}/2=9/2
{2x^2}/2+{8x}/2=9/2
x^2+4x=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.
(4/2)^2
(2)^2
4
x^2+4x=9/2
x^2+4x+4=9/2+4
The left hand side factors to be a perfect square
x^2+4x+4=9/2+4
(x+2)(x+2)=9/2+4
(x+2)^2=9/2+4
Simplify the right hand side by finding a common denominator and adding the fractions.
(x+2)^2=9/2+4
(x+2)^2=9/2+4/1
(x+2)^2=9/2+{4*2}/{1*2}
(x+2)^2=9/2+8/2
(x+2)^2=17/2
Solve the equation using the square root method.  Start by taking the square root of both sides to get rid of the square
(x+2)^2=17/2
sqrt{(x+2)^2}=sqrt{17/2}
x+2=pm sqrt{17/2}
Solve for x by subtracting 2 from both sides.
x+2=pm sqrt{17/2}
x+2-2=-2pm sqrt{17/2}
x=-2pm sqrt{17/2}
Rationalize the denominator by applying the radical to the numerator and denominator
x=-2pm sqrt{17/2}
x=-2pm sqrt{17}/sqrt{2}
Rationalize the denominator by multiplying by the fraction  sqrt{2}/sqrt{2}
x=-2pm sqrt{17}/sqrt{2}
x=-2pm sqrt{17}/sqrt{2} sqrt{2}/sqrt{2}
x=-2pm sqrt{17}/sqrt{2}
x=-2pm sqrt{34}/sqrt{4}
x=-2pm sqrt{34}/2