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Solving Quadratic Equations
Quadratic Equation
An equation in which one or more of the terms is squared but raised to no higher
power, having the general form ax2 + bx + c = 0, where a, b, and c are
constants. a is the leading coefficient
Below are examples of different methods of solving quadratic equations.
Solve by Taking Square Root
If the quadratic equation only has an x2 term and a constant, you can solve by
moving the constant to one side of the equation and the squared variable to the
other. Then take the square root of both sides.
Example
Solve for x: x2 – 9 = 0.
Steps
1. Add 9 to both sides
x2 = 9
2. Take square root of both sides
x2 = 9
3. Simplify radical
x = 3, -3
So, x = 3, -3 are the solutions to the equation.
Solve by Factoring
If there is also an x term in the quadratic, you may be able to solve by factoring.
Below is an example of a quadratic equation that is solved by factoring. Please
watch the video tutorial to see this problem explained more thoroughly.
Example
Solve for x: x2 – x – 6 = 0.
Steps
1. Factor quadratic into linear factors
(x – 3)(x + 2)
2. Set each factor = 0
x–3=0;x+2=0
3. Solve each equation
x = 3 ; x = -2
So, x = -2, 3 are the solutions to the equation.
Solve by Quadratic Formula
Sometimes you will not be able to factor a quadratic equation. So, the quadratic
formula can be used instead. In fact, the quadratic formula is the only method
that will work in solving any quadratic equation. The quadratic formula states
that for any quadratic equation in the form of ax2 + bx + c = 0, you can solve
for x by using the following formula:
− b ± b 2 − 4ac
x=
2a
© LaurusSoft, Inc.
The following is an example of how to solve a quadratic equation using the
quadratic formula.
Example
Solve for x: x2 – x = 3
Steps
1. Put equation in standard form (subtract 3) 1x2 – 1x – 3 = 0
2. Find values of a, b, and c
a = 1, b = -1, c = -3
3. Plug a, b, and c into quadratic formula
− (−1) ± (−1) 2 − 4(1)(-3)
x=
2(1)
4. Simplify
x=
So, the solutions of this equation are x =
1 ± 1 + 12 1 ± 13
=
2
2
1 + 13
1 − 13
and x =
2
2
Solve Completing the Square
A quadratic equation in the form ax2 + bx + c = 0, where a = 1 and a, b, and c
are real numbers, can be written as a perfect square trinomial in the form
2
⎛b⎞
⎛b⎞
x 2 + bx + ⎜ ⎟ = −c + ⎜ ⎟
⎝2⎠
⎝2⎠
2
and factored as
⎛
⎛ b ⎞⎞
⎛b⎞
⎜⎜ x − ⎜ ⎟ ⎟⎟ = −c + ⎜ ⎟
⎝ 2 ⎠⎠
⎝2⎠
⎝
2
2
Now you can solve by taking the square roots of both sides.
Example
Solve x2 – 6x – 7 = 0.
1. Set equation = 0. (done)
2. Move constant to other side. (add 7)
⎛b⎞
3. Write linear binomial (x – ⎜ ⎟ )2 (b = 6)
⎝2⎠
x2 – 6x – 7 = 0
x2 – 6x
=7
(x – 3)2 Æ x2 – 6x + 9
2
4.
5.
6.
7.
⎛b⎞
Add ⎜ ⎟ to both sides. (add 9)
⎝2⎠
Simplify
Take square root of both sides
Solve for x (add 3 to both sides)
© LaurusSoft, Inc.
(x – 3)2 = 7 + 9
(x – 3)2 = 16
x–3=4
x=7