10.1 Solve Quadratic Equations Using the Square Root Property
- Square Root Property
If , and , then .
10.2 Solve Quadratic Equations by Completing the Square
- Binomial Squares Pattern If are real numbers,
- Complete a Square
To complete the square of :
- Step 1. Identify , the coefficient of .
- Step 2. Find , the number to complete the square.
- Step 3. Add the to .
10.3 Solve Quadratic Equations Using the Quadratic Formula
- Quadratic Formula The solutions to a quadratic equation of the form are given by the formula:
- Solve a Quadratic Equation Using the Quadratic Formula
To solve a quadratic equation using the Quadratic Formula.
- Step 1. Write the quadratic formula in standard form. Identify the values.
- Step 2. Write the quadratic formula. Then substitute in the values of
- Step 3. Simplify.
- Step 4. Check the solutions.
- Using the Discriminant, , to Determine the Number of Solutions of a Quadratic Equation
For a quadratic equation of the form
- if , the equation has 2 solutions.
- if , the equation has 1 solution.
- if , the equation has no real solutions.
- To identify the most appropriate method to solve a quadratic equation:
- Step 1. Try Factoring first. If the quadratic factors easily this method is very quick.
- Step 2. Try the Square Root Property next. If the equation fits the form or , it can easily be solved by using the Square Root Property.
- Step 3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.
10.4 Solve Applications Modeled by Quadratic Equations
- Area of a Triangle For a triangle with base, , and height, , the area, , is given by the formula:
- Pythagorean Theorem In any right triangle, where and are the lengths of the legs, and is the length of the hypothenuse,
- Projectile motion The height in feet, , of an object shot upwards into the air with initial velocity, , after seconds can be modeled by the formula:
10.5 Graphing Quadratic Equations
- The graph of every quadratic equation is a parabola.
- Parabola Orientation For the quadratic equation , if
- , the parabola opens upward.
- , the parabola opens downward.
- Axis of Symmetry and Vertex of a Parabola For a parabola with equation :
- The axis of symmetry of a parabola is the line .
- The vertex is on the axis of symmetry, so its x-coordinate is .
- To find the y-coordinate of the vertex we substitute into the quadratic equation.
- Find the Intercepts of a Parabola To find the intercepts of a parabola with equation :
- To Graph a Quadratic Equation in Two Variables
- Step 1. Write the quadratic equation with on one side.
- Step 2. Determine whether the parabola opens upward or downward.
- Step 3. Find the axis of symmetry.
- Step 4. Find the vertex.
- Step 5. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
- Step 6. Find the x-intercepts.
- Step 7. Graph the parabola.
- Minimum or Maximum Values of a Quadratic Equation
- The y-coordinate of the vertex of the graph of a quadratic equation is the
- minimum value of the quadratic equation if the parabola opens upward.
- maximum value of the quadratic equation if the parabola opens downward.