Steps for solving quadratic equations by factoring:
There are four steps used for solving quadratic equations by factoring.
Step 1. Using the addition & subtraction axioms, arrange an equation in the common quadratic form ax2 + bx + c = 0.
Step 2. Factor the left-hand side of the equation.
Step 3. Set each factor equal to zero and solve the resulting linear equations.
Step 4. Check the roots.
Example:
Solve the subsequent quadratic equation by factoring.
2x2 - 3 = 4x - x2 + 1
Solution:
Step 1. Using the subtraction axiom, subtract (4x - x2 + 1) from both sides of the equation.
2x2 - 3 - (4x - x2 + 1) = 4x - x2 + 1 - (4x - x2 + 1)
3x2 - 4x - 4 = 0
Step 2. Factor the resulting equation.
3x2 - 4x - 4 = 0
(3x + 2) (x - 2) = 0
Step 3. Set each factor equal to zero and solve the resulting equations.
3x + 2 = 0
3x = -2
3x/3 = -2/3
X= -2/3
x - 2 = 0
x = 2
Thus, the roots are x = -2/3 and x =2.
Step 4. Check the roots.
2x2- 3 = 4x -x2+1
2(-2/3)2 - 3 = 4 (-2/3)- (-2/3)2 +1
2(4/9) - 3 = -(8/3) - (4/9) +1
(8/9) - (27/9) = -(24/9) - (4/9) + (9/9)
-(19/9) = -(19/9)
2x2 - 3 = 4x - x2 + 1
2(2)2 - 3 = 4(2) - (2)2 +1
2(4) - 3 = 8 - 4 + 1
8 - 3 = 5
5 = 5
Therefore, the roots check.
Quadratic equations in that the numerical constant c is zero can always be solved through factoring. One of the two roots is zero. For instance, the quadratic equation 2x2 + 3x = 0 can be solved through factoring. The factors are (x) and (2x + 3). Therefore, the roots are x = 0 and x = - 3/2 . If a quadratic equation in that the numerical constant c is zero is written in common form, a general expression can be written for its roots. A general form of a quadratic equation in that the numerical constant c is zero is the subsequent:
ax2 + bx = 0
In the above equation the left-hand side can be factored through removing an x from each term.
x(ax + b) = 0
In the given equation the roots of this quadratic equation are found by setting the two factors equal to zero and solving the resulting equations.
x = 0
x = - b /a
Therefore, the roots of a quadratic equation in which the numerical constant c is zero are x = 0 and x = - b/a .