Section 4.2 Notes Solving Quadratic Equations by Graphing

Quadratic equations are quadratic functions that are set equal to a value. The standard form of a quadratic equation is ax2 + bx + c = 0, where a ≠ 0 and a, b, and c are integers. The solutions of a quadratic equation are called the roots of the equation. One method for finding the roots of a quadratic equation is to find the zeros of the related quadratic function. Root, zero, and solution all mean the x-intercepts of a function, or the values that make f(x) = 0.

Steps to Solve by Graphing 1. Put the equation into Standard form (ax2 + bx + c = 0) 2. Replace the 0 with a y. So you will have y = ax2 + bx + c 3. Graph the equation 4. Look at the x-intercepts to find the solution 5. Check your solution Example 1: Solve x2 + 6x + 8 = 0 by graphing. x f(x)  

Example 2: Solve x2 + 2x – 3 = 0 by graphing. a) b) c) d)

Example 3: Solve x2 – 4x = –4 by graphing f(x)  

Example 4: Use a quadratic equation to find two numbers with a sum of 4 and a product of 5.

Example 5: Solve –x2 + 4x – 1 = 0 by graphing on the calculator Example 5: Solve –x2 + 4x – 1 = 0 by graphing on the calculator. If exact roots cannot be found, state the consecutive integers between which the roots are located.

Example 6: Solve x2 + 5x – 7 = 0 by graphing on the calculator Example 6: Solve x2 + 5x – 7 = 0 by graphing on the calculator. If exact roots cannot be found, state the consecutive integers between which the roots are located.

Example 7: The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053 feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h(t) = –16t 2 + h0, where t is the time in seconds and h0 is the initial height above the water in feet.