- The Quadratic Formula
- Given the problem
Solve ax2 + bx + c = 0,
the solutions are
- The quadratic formula is easily derived by completing the square
to solve the equation ax2 + bx + c = 0.
- Using the Quadratic Formula
- Write the given quadratic equation in standard form ax2 + bx + c = 0.
- Determine the numerical values for a, b, and c.
- Substitute the values for a, b, and c into the quadratic formula.
- Evaluate the formula to obtain the solutions.
- The Discriminant, b2 - 4ac
- The discriminant tells how many different values will be solutions.
- If b2 - 4ac > 0,
- the quadratic equation has 2 distinct real number solutions.
- If b2 - 4ac = 0,
- the quadratic equation has a single real number solution (repeated twice).
- If b2 - 4ac < 0,
- the quadratic equation has no real number solutions, it has 2 complex solutions.
- Sketching Parabolas
A parabola is the name of a graph with the equation
y = ax2 + bx + c.
- Steps to sketching the graph of a parabola
- Determine whether the parabola opens upward or downward:
- If a > 0, the parabola opens upward;
- If a < 0, the parabola opens downward.
- Find the y-intercept:
- Find the value of y when x = 0.
- Find the x-intercepts:
- The x-intercepts are the points where y = 0.
- So, solve ax2 + bx + c = 0.
- The solutions are the x-coordinates of the x-intercepts.
- There may be 0, 1, or 2 x-intercepts.
- Find the vertex of the parabola
- The x-coordinate of the vertex is x = -b/2a.
- To find the y-coordinate of the vertex, plug the number x = -b/2a into y = ax2 + bx + c.
- Plot the y-intercept, the x-intercepts (if any), and the vertex.
- Sketch the graph by drawing a smooth curve, with the shape of parabola, through the points just plotted and opening in the direction found in step 1.