- This section covers using factoring to algebraically solve for x equations having the form
ax2 + bx + c = 0, a not zero,
which is the standard form of a quadratic equation in one variable.
- Zero-Factor Property
- For all real-valued factors a and b
- if a · b = 0,
- then one (or both) of a and b must be 0.
- To Solve a Quadratic Equation in One Variable by Factoring
- Get 0 on one side of the "=" sign and everything else on the other side.
- Simplify by combining like terms (on the side of the equation without the 0).
- Factor (the side of the equation without the 0).
- Set to 0 each factor containing the variable.
(This is application of the zero-factor property.)
- To determine the solutions, solve each equation obtained in step 4.
- Check the solutions in the original equation.
- All quadratic equations have 2 solutions.
- Sometimes the 2 solutions have the same value.
- Not all quadratic equations have real-valued solutions.
- Finding the Intercepts of the Graph of a Quadratic Equation
- Considering the graph of y = ax2 + bx + c,
- To find the x-intercepts
- Set y = 0 and solve for x.
- That means, solve 0 = ax2 + bx + c.
- The x-intercepts are (x1, 0) and (x2, 0),
if the solutions are called x1 and x2.
- To find the y-intercept
- Set x = 0 and solve for y.
- That means, y = c.
- The y-intercept is (0, c).