Solving Quadratic Equations
A quadratic equation is an equation of the form ax2+bx+c=0.
For example, 2x2-7x-15=0.
So, what does it mean to solve a quadratic equation?
Solving a quadratic equation means finding those values of the variable that make the equation true.
For example, solving 6x2+10x+3=7 means finding the values of x that make this equation true.
If x=-2, 6(-2)2+10(-2)+3=7
Only these two values of x: -2 and 1/3 make this equation true, so these are the solutions to the equation 6x2+10x+3=7.
How do I find these values of x that make the equation true?
Before we discuss that, there is an important theorem we need to revisit:
If axb=0 then
- or both a and b are zero.
Now we have reviewed the Zero Factor Principle, we can continue our exploration into solving quadratic equations. We see now that we can solve any factor-equation of the form ab=0.
We will look at a specific example of a factorable quadratic equation 6x2+x-15=0 where a = 6, b = 1 and c = -15. Note:the sign on the number must be included.
|Step 1: Multiply a by c.||a x c = ac||6 x -15 = -90|
|Step 2: Find the factors of ac that add to give b.||-90 = -9 x 10; -9 + 10 = 1|
|Step 3: Rewrite the equation, represent b as the
sum of the two factors you just found.
|Step 4: Factor the equation by grouping.||3x(2x-3)+5(2x-3)=0|
|This equation now has two terms that
have a common factor of (2x-3), factor again.
Now you have a factor-equation of the form a x b = 0: (2x-3)(3x+5)=0
Here we will use the Zero Factor Principle, so
- 2x-3 = 0, and so solving for x, x = 3/2
- 3x+5 = 0, and so solving for x, x = -5/3
Solve the following quadratic equations, using the method shown above:
Step 1: 1 x -3 = -3
Step 2: Factors of -3 that add to -2: 1 and -3
Step 3: x2+x-3x-3=0
Step 4: x(x+1)-3(x+1)=0
x+1=0 and x-3=0
Therefore, x = -1 and 3