One way to solve the equation \begin{align*}x^22x3=0\end{align*}
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James Sousa: Solve a Quadratic Equation Graphically on the Calculator
Guidance
Recall that a quadratic equation is a degree 2 equation that can be written in the form \begin{align*}ax^2+bx+c=0\end{align*}
How do the solutions to the equation \begin{align*}x^2+x12=0\end{align*}
You can see the xintercepts are at \begin{align*}(4,0)\end{align*}

\begin{align*}(4)^2+(4)12=16412=0\end{align*}
(−4)2+(−4)−12=16−4−12=0

\begin{align*}(3)^2+(3)12=9+312=0\end{align*}
(3)2+(3)−12=9+3−12=0
Graphing is a great way to solve quadratic equations. Keep in mind that you can also solve many quadratic equations by factoring or using other algebraic methods such as the quadratic formula or completing the square.
Example A
Solve the following quadratic equation by finding the xintercepts of the corresponding quadratic function: \begin{align*}x^22x8=0\end{align*}
Solution: The corresponding function is \begin{align*}y=x^22x8\end{align*}
The \begin{align*}x\end{align*}
Example B
Solve the following quadratic equation by finding the xintercepts of the corresponding quadratic function: \begin{align*}x^2+4x+4=0\end{align*}
Solution: The corresponding function is \begin{align*}y=x^2+4x+4\end{align*}
The only \begin{align*}x\end{align*}
Example C
Solve the following quadratic equation by finding the xintercepts of the corresponding quadratic function: \begin{align*}x^2+3x=10\end{align*}
Solution: First rewrite the equation so it is set equal to zero to get \begin{align*}x^2+3x10=0\end{align*}
For this example you will see how the calculator can calculate the zeros of a function on a graph. This technique is particularly useful when the intercepts are not at whole numbers. Have the calculator find the \begin{align*}x\end{align*}
The calculator will display “Left Bound?” Use the arrow to position the cursor so that it is to the left and above the \begin{align*}x\end{align*}
When the cursor has been positioned, press
The calculator will now display “Right Bound?” Use the arrows to position the cursor so that it is to the right and below the \begin{align*}x\end{align*}
When the cursor has been positioned, press
The calculator will now display “Guess?”
Press
At the bottom of the screen you can see it says "Zero" and the x and y coordinates. You are interested in the xcoordinate because that is one of the solutions to the original equation. The \begin{align*}x\end{align*}
Repeat this same process to determine the value of the \begin{align*}x\end{align*}
The \begin{align*}x\end{align*}
Concept Problem Revisited
To solve the equation \begin{align*}x^22x3=0\end{align*}
Another way to think about this problem is to solve the system:
\begin{align*}\begin{Bmatrix} y= x^2 2x 3 \\ y= 0 \end{Bmatrix}\end{align*}
You are looking for where the parabola \begin{align*}y=x^22x3\end{align*} intersects with the line \begin{align*}y=0\end{align*}.
The points of intersection are (–1, 0) and (3, 0). The solutions to the original equation are \begin{align*}x=1\end{align*} and \begin{align*}x=3\end{align*}.
Vocabulary
 Quadratic Equation
 A quadratic equation is an equation of degree 2. The standard form of a quadratic equation is \begin{align*}ax^2+bx+c=0\end{align*} where \begin{align*}a\ne 0\end{align*}.
 Quadratic Function
 A quadratic function is a function that can be written in the form \begin{align*}f(x)=ax^2+bx+c\end{align*} with \begin{align*}a\ne 0\end{align*}. The graph of a quadratic function is a parabola.
 Zeros of a Quadratic Function
 The zeros of a quadratic function are the \begin{align*}x\end{align*}intercepts of the function. These are the values for the variable ‘\begin{align*}x\end{align*}’ that will result in \begin{align*}y = 0\end{align*}.
 Roots of a Quadratic Function
 The roots of a quadratic function are also the \begin{align*}x\end{align*}intercepts of the function. These are the values for the variable ‘\begin{align*}x\end{align*}’ that will result in \begin{align*}y = 0\end{align*}.
Guided Practice
Solve each quadratic equation using a graph.
1. \begin{align*}x^23x10=0\end{align*}
2. \begin{align*}2x^25x+2=0\end{align*}
3. \begin{align*}2x^25x=3\end{align*}
Answers:
1. To begin, create a table of values for the corresponding function \begin{align*}y=x^23x10\end{align*} by using your graphing calculator:
 From the table, the \begin{align*}x\end{align*}intercepts are (–2, 0) and (5, 0). The \begin{align*}x\end{align*}intercepts are the values for ‘\begin{align*}x\end{align*}’ that result in \begin{align*}y = 0\end{align*} and are therefore the solutions to the equation.
 The solutions to the equation are are \begin{align*}x = 2\end{align*} and \begin{align*}x = 5\end{align*}.
2. To begin, create a table of values for the corresponding function \begin{align*}y=2x^25x+2\end{align*} by using your graphing calculator:
 Press
 Press the following keys to determine the \begin{align*}x\end{align*}intercept to the left:
 Press the following keys to determine the \begin{align*}x\end{align*}intercept to the right:
 The \begin{align*}x\end{align*}intercepts of the function are (0.5, 0) and (2, 0). The solutions to the equation are, therefore, \begin{align*}x = 0.5\end{align*} and \begin{align*}x = 2\end{align*}.
3. First rewrite the equation so it is set equal to zero: \begin{align*}2x^25x3=0\end{align*}. Next, create a table of values for the corresponding function \begin{align*}y=2x^25x3\end{align*} by using your graphing calculator:
 Now sketch the graph of the function.
 The zeros of the function are (–0.5, 0) and (3, 0). Therefore, the solutions to the equation are \begin{align*}x = 0.5\end{align*} and \begin{align*}x = 3\end{align*}.
Practice
Use your graphing calculator to solve each of the following quadratic equations by graphing:
 \begin{align*}2x^2+9x18=0\end{align*}
 \begin{align*}3x^2+8x3=0\end{align*}
 \begin{align*}5x^2+13x+6=0\end{align*}
 \begin{align*}2x^211x+5=0\end{align*}
 \begin{align*}3x^2+8x3=0\end{align*}
 \begin{align*}x^2x20=0\end{align*}
 \begin{align*}2x^27x+5=0\end{align*}
 \begin{align*}3x^2+7x=2\end{align*}
 \begin{align*}2x^215=x\end{align*}
 \begin{align*}3x^210x=8\end{align*}
 How could you use the graphs of a system of equations to solve \begin{align*}3x^210x=8\end{align*}?
 What's the difference between a quadratic equation and a quadratic function?
 Will a quadratic equation always have 2 solutions? Explain.
 The quadratic equation\begin{align*}x^2+4=0\end{align*} has no real solutions. How does the graph of \begin{align*}y=x^2+4\end{align*} verify this fact?
 When does it make sense to use the graphing method for solving a quadratic equation?