9.4 – Solving Quadratic Equations

BY GRAPHING!

Warm-Up

1. Solve for x in the equation

2. Graph

What is a Quadratic Equation?

A quadratic equation in standard form is written:

y = ax2 + bx + c, where a ≠ 0

Roots of a Quadratic Equation

• The roots of a quadratic equation are the solutions to:

0 = ax2 + bx + c

Quadratic equation in standard formwith y = 0

What kind of points on a graph have y-values of 0? Where do we find these points? What might we call them?

Roots of a Quadratic Equation

• Roots are represented graphically by the x-intercepts of the graph of a quadratic equation.

RootsRoots, Roots, Baby!

Connecting Solutions to Roots

Let’s look at the equation:

Solve for .

Connecting Solutions to Roots

Without doing any calculations solve , given the graph below of .

x = 2, -2

Quick Practice!

Put the following equation into standard form:

Quick Practice!

Below is the graph of , determine the roots and check them algebraically.

Quick Practice!

Put the following equation into standard form:

Quick Practice!

Below is the graph of , determine the roots and check them algebraically.

Quick Practice!

Put the following equation into standard form:

Quick Practice!

Below is the graph of , determine the roots and check them algebraically.

So how does this help me?

Using graphing to solve:

1. Rewrite the equation into standard form.2. Change the 0 into a y and graph the

equation: .3. Identify the roots of the equation, which are

your solutions to the original equation.4. Check to see if these work algebraically!

I GET IT NOW!!!

Getting it Done by Hand

Solve the following equation by graphing (you may not use any graphing technology):

𝑥2−𝑥=2

x -1 0 1 2

y

Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=

button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),

setting left and right boundaries and making a guess. 5. Check answers!

Graph: 4x2 = 16

Using a Calculator

Using a Calculator

Graph: x2 - 4x = 5

Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=

button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),

setting left and right boundaries and making a guess. 5. Check answers!

Using a Calculator

Graph: x2 = -x + 6

Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=

button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),

setting left and right boundaries and making a guess. 5. Check answers!

Using a Calculator

Graph: Make one up!

Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=

button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),

setting left and right boundaries and making a guess. 5. Check answers!

A baseball is thrown at 100 mph ft/sec from left field toward home plate. The models below give paths of the ball for two initial angles, with height of y and horizontal distance x (both measure in feet)

o 2

o 2

15 : y .00171x .268x 6

25 : y .00195x .466x 6

If home plate is 236 feet away, which angle(s) have the ball hitting the ground before reaching the plate?

Homework

Complete worksheet by tomorrow!

Quiz 9.4-9.6 on Monday 5/6!