1. Solving Quadratic Equations by Factoring
The general form of a quadratic equation is
ax2 + bx + c = 0
where x is the variable and a, b & c are constants
Examples of Quadratic Equations
(a) 5x2 − 3x − 1 = 0 is a quadratic equation in quadratic form where
`a = 5`, `b = -3`, `c = -1`
(b) 5 + 3t − 4.9t2 = 0 is a quadratic equation in quadratic form.
Here, `a = -4.9`, `b = 3`, `c = 5`
[This equation arose from finding the time when a projectile, being acted on by gravity, hits the ground.]
(c) (x + 1)2 = 4 is a quadratic equation but not in quadratic form.
It has to be expanded and simplified to:
x2 + 2x − 3 = 0
Summary
In general, a quadratic equation:
- must contain an x2 term
- must NOT contain terms with degrees higher than x2 eg. x3, x4 etc
Examples of NON-quadratic Equations
- bx − 6 = 0 is NOT a quadratic equation because there is no x2 term.
- x3 − x2 − 5 = 0 is NOT a quadratic equation because there is an x3 term (not allowed in quadratic equations).
Solutions of a Quadratic Equation
The solution of an equation consists of all numbers (roots) which make the equation true.
All quadratic equations have 2 solutions (ie. 2 roots). They can be:
- real and distinct
- real and equal
- imaginary (complex)
Example 1
The quadratic equation x2 − 7x + 10 = 0 has roots of
`x = 2` and `x = 5`. (We'll show below how to find these roots.)
This can be seen by substituting in the equation:
When x = 2,
x2 − 7x + 10
= (2)2 − 7(2) + 10
= 4 − 14 + 10
= 0
(This can be shown similarly for x = 5). In this example, the roots are real and distinct.
Example 2
The quadratic equation x2 − 6 x + 9 = 0 has double roots of x = 3 (both roots are the same)
This can be seen by substituting x = 3 in the equation:
x2 − 6x + 9
= (3)2 − 6(3) + 9
= 9 − 18 + 9
= 0
Example 3
The quadratic equation
x2 + 9 = 0
has imaginary roots of
`x=sqrt(-9)` or `-sqrt(-9)`
Learn more about imaginary numbers.
Solving a Quadratic Equation by Factoring
For the time being, we shall deal only with quadratic equations that can be factored (factorised).
If you need a reminder on how to factor, go back to the section on:
Factoring Trinomials.
Using the fact that a product is zero if any of its factors is zero we follow these steps:
(i) Bring all terms to the left and simplify, leaving zero on the right side.
(ii) Factorise the quadratic expression
(iii) Set each factor equal to zero
(iv) Solve the resulting linear equations
(v) Check the solutions in the original equation
Example 4
Solve x2 − 2x − 15 = 0
Answer
x2 − 2x − 15 = 0
Factoring gives:
(x − 5)(x + 3) = 0
Now, if either of the terms (x − 5) or (x + 3) is 0, the product is zero. So we conclude:
(x − 5) = 0, therefore
x = 5
or
(x + 3) = 0, therefore
x = − 3
Hence the roots are x = 5 and x = − 3.
Are we correct?
We check the roots in the original equation by substitution.
When x = 5:
x2 − 2x − 15
= (5)2 − 10 − 15
= 25 − 10 − 15
= 0
(Similarly, when we substitute `x = -3`, we also get `0`.)
Alternate method (Po-Shen Loh's approach)
We could have proceded as follows to solve this quadratic equation. The following approach takes the guesswork out of the factoring step, and is similar to what we'll be doing next, in Completing the Square.
Step 1: Take −1/2 times the x coefficient. In this case, `-1/2 × (−2) = 1`
Step 2: Expand `(1 − u)(1 + u) = 1 − u^2`
Step 3: Set that expansion equal to the constant term: `1 - u^2 = -15`
Step 4: Solve for `u`:
`1 - u^2 = -15`
`u^2 = 16`
`u= +-4`
Step 5: Substitute either value (we'll use `+4`) into the `u` bracket expressions, giving us the same roots of the quadratic equation that we found above:
`x=(1-u)=1-4 = -3,` or
`x=(1+u)=1+4 = 5`
For more on this approach, see: A Different Way to Solve Quadratic Equations (video by Po-Shen Loh).
Example 5
Solve
`9x^2+ 6x + 1 = 0`
Answer
9x2 + 6x + 1 = 0
Factoring gives:
(3x + 1)(3x + 1) = 0
So we conclude:
(3x + 1) = 0,
therefore
`x=-1/3`
We say there is a double root of `x = -1/3`.
Alternate method (Po-Shen Loh's approach)
We first need to re-express the quadratic with a `1` as the coefficient of `x^2.`
`x^2+ 2/3x + 1/9 = 0`
Step 1: Take −1/2 times the x coefficient. In this case, `-1/2 × 2/3 = -1/3`
Step 2: Expand `(-1/3 − u)(-1/3 + u) = 1/9 − u^2`
Step 3: Set that expansion equal to the constant term: `1/9 - u^2 = 1/9`
Step 4: Solve for `u`:
`1/9 - u^2 = 1/9`
`u^2 = 0`
`u= 0`
Step 5: Substitute `u=0` into the `u` bracket expressions, giving us the same (repeated) root for the quadratic equation that we found above:
`x=-1/3-0 = -1/3,` or `x=-1/3+0 = -1/3`
Example 6 (involving fractions)
Solve
`2-1/x=3/(x+2)`
Answer
`2-1/x=3/(x+2)`
Multiply throughout by `x(x+2)` to remove the denominators (bottoms) of the fractions:
`2x(x+2)-(x(x+2))/x=(3(x)(x+2))/(x+2)`
Cancelling gives:
`2x(x+2)-(x+2)=3x`
Expanding the brackets:
`2x^2+4x-x-2=3x`
`2x^2-2=0`
`x^2-1=0`
Factoring gives:
`(x+1)(x-1)=0`
So `x = -1` or `x = 1`.
CHECK: Substituting `x = -1` into both the left hand side and right hand side of the question gives:
`"LHS"=2-1/x=2-1/-1=3`
`"RHS"=3/(x+2)=3/(-1+2)=3="LHS" `
Likewise, for `x = +1`,
LHS `= 2 - 1 = 1`
RHS `= 3/3 = 1 =` LHS
Exercises
- Determine if the following are quadratic equations. If so, determine a, b, and c.
a. 5x2 = 9 − x
b. (3x − 2)2 = 2
Answer
Q1a `5x^2= 9 − x`
`5x^2+ x − 9 = 0`
So, yes, it is a quadratic equation with
`a = 5`, `b = 1`, `c = -9`
Q1b `(3x − 2)^2= 2`
`9x^2− 12x + 4 = 2`
`9x^2− 12x + 2 = 0`
So, yes, it is a quadratic equation with
`a = 9`, `b = -12`, `c = 2`
- Solve for x:
2x2 − 7x + 6 = 3
Answer
2x2 − 7x + 6 = 3
2x2 − 7x + 3 = 0
(2x − 1)(x − 3) = 0
So
`x=1/2` or `x=3`.
Alternate method (Po-Shen Loh's approach)
Once again, we first need to re-express the quadratic with a `1` as the coefficient of `x^2.`
`x^2- 7/2x + 3/2 = 0`
Step 1: Take −1/2 times the x coefficient. In this case, `-1/2 × (-7/2) = 7/4`
Step 2: Expand `(7/4 − u)(7/4 + u) = 49/16 − u^2`
Step 3: Set that expansion equal to the constant term: `49/16 - u^2 = 3/2`
Step 4: Solve for `u`:
`49/16 - u^2 = 3/2`
`u^2 = 49/16 - 24/16 = 25/16`
`u= +-5/4`
Step 5: Substitute either value (we'll use `+5/4`) into the `u` bracket expressions, giving us the same roots of the quadratic equation that we found above:
`x=7/4-5/4 = 1/2,` or `x=7/4+5/4 = 3`