Math 101 Intermediate Algebra

Solving Quadratic Equations by the Quadratic Formula
Chapter 9, Section 2

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

  1. Write the given quadratic equation in standard form ax2 + bx + c = 0.

  2. Determine the numerical values for a, b, and c.

  3. Substitute the values for a, b, and c into the quadratic formula.

  4. 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

  1. Determine whether the parabola opens upward or downward:
    • If a > 0, the parabola opens upward;
    • If a < 0, the parabola opens downward.

  2. Find the y-intercept:
    • Find the value of y when x = 0.

  3. 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.

  4. 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.

  5. Plot the y-intercept, the x-intercepts (if any), and the vertex.

  6. 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.