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# Use Graphs to Solve Quadratic Equations

## Identify x-intercepts of parabolas

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Practice Use Graphs to Solve Quadratic Equations
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One way to solve the equation x22x3=0\begin{align*}x^2-2x-3=0\end{align*} is to use factoring and the zero product property. How could you use a graph to solve x22x3=0\begin{align*}x^2-2x-3=0\end{align*}?

### Guidance

Recall that a quadratic equation is a degree 2 equation that can be written in the form ax2+bx+c=0\begin{align*}ax^2+bx+c=0\end{align*}. Every quadratic equation has a corresponding quadratic function that you get by changing the "0\begin{align*}0\end{align*}" to a "y\begin{align*}y\end{align*}". Standard form for a quadratic function is y=ax2+bx+c\begin{align*}y=ax^2+bx+c\end{align*}. Quadratic functions can be graphed by hand, or with a graphing calculator.

How do the solutions to the equation x2+x12=0\begin{align*}x^2+x-12=0\end{align*} show up on the graph of y=x2+x12\begin{align*}y=x^2+x-12\end{align*}? On the graph you are looking for the points that have a y-coordinate that is equal to 0. Therefore, the solutions to the equation will show up as the x-intercepts on the graph of the function. These are also known as the roots or zeros of the function. Here is the graph of y=x2+x12\begin{align*}y=x^2+x-12\end{align*}:

You can see the x-intercepts are at (4,0)\begin{align*}(-4,0)\end{align*} and (3,0)\begin{align*}(3,0)\end{align*}. This means that the solutions to the equation x2+x12=0\begin{align*}x^2+x-12=0\end{align*} are x=4\begin{align*}x=-4\end{align*} and x=3\begin{align*}x=3\end{align*}. You can verify these solutions by substituting them back into the equation:

• (4)2+(4)12=16412=0\begin{align*}(-4)^2+(-4)-12=16-4-12=0\end{align*}
• (3)2+(3)12=9+312=0\begin{align*}(3)^2+(3)-12=9+3-12=0\end{align*}

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 x-intercepts of the corresponding quadratic function: x22x8=0\begin{align*}x^2-2x-8=0\end{align*}

Solution: The corresponding function is y=x22x8\begin{align*}y=x^2-2x-8\end{align*}. Use your graphing calculator to make a table and a graph for this function.

The x\begin{align*}x\end{align*}-intercepts are (–2, 0) and (4, 0). The x\begin{align*}x\end{align*}-intercepts are the values for ‘x\begin{align*}x\end{align*}’ that result in y=0\begin{align*}y = 0\end{align*} and are therefore the solutions to your equation. The solutions for the quadratic are x=2\begin{align*}x = -2\end{align*} and x=4\begin{align*}x = 4\end{align*}.

#### Example B

Solve the following quadratic equation by finding the x-intercepts of the corresponding quadratic function: x2+4x+4=0\begin{align*}x^2+4x+4=0\end{align*}

Solution: The corresponding function is y=x2+4x+4\begin{align*}y=x^2+4x+4\end{align*}. Use your graphing calculator to make a table and a graph for this function.

The only x\begin{align*}x\end{align*}-intercept is (–2, 0). There is only one solution to the equation: x=2\begin{align*}x=-2\end{align*}. Keep in mind that quadratic equations can have 0, 1, or 2 real solutions. If you were to factor the quadratic x2+4x+4\begin{align*}x^2+4x+4\end{align*}, you would get (x+2)(x+2)\begin{align*}(x+2)(x+2)\end{align*}--two of the same factors. The root of 2\begin{align*}-2\end{align*} for this function is said to have a multiplicity of 2, because 2 factors produce the same solution. You will learn more about multiplicity when you study polynomials in future courses.

#### Example C

Solve the following quadratic equation by finding the x-intercepts of the corresponding quadratic function: x2+3x=10\begin{align*}x^2+3x=10\end{align*}

Solution: First rewrite the equation so it is set equal to zero to get x2+3x10=0\begin{align*}x^2+3x-10=0\end{align*}. Now, the corresponding function is y=x2+3x10\begin{align*}y=x^2+3x-10\end{align*}. Use your graphing calculator to make a graph for this function. You will see that there are two x\begin{align*}x\end{align*}-intercepts.

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 x\begin{align*}x\end{align*}-intercept on the left first. Press

The calculator will display “Left Bound?” Use the arrow to position the cursor so that it is to the left and above the x\begin{align*}x\end{align*}-axis.

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 x\begin{align*}x\end{align*}-axis.

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 x-coordinate because that is one of the solutions to the original equation. The x\begin{align*}x\end{align*}-intercept is (–5, 0) which means that one of the solutions is x=5\begin{align*}x = - 5\end{align*}.

Repeat this same process to determine the value of the x\begin{align*}x\end{align*}-intercept on the right.

The x\begin{align*}x\end{align*}-intercept is (2, 0) which means that the second solution is x=2\begin{align*}x = 2\end{align*}.

#### Concept Problem Revisited

To solve the equation x22x3=0\begin{align*}x^2-2x-3=0\end{align*} using a graph, use a calculator to graph the corresponding function y=x22x3\begin{align*}y=x^2-2x-3\end{align*}. Then, look for the values on the graph where y=0\begin{align*}y=0\end{align*}, which will be the x-intercepts.

You are looking for where the parabola \begin{align*}y=x^2-2x-3\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

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*}.
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.
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*}.
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.

2.

3.

1. To begin, create a table of values for the corresponding function \begin{align*}y=x^2-3x-10\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^2-5x+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^2-5x-3=0\end{align*}. Next, create a table of values for the corresponding function \begin{align*}y=2x^2-5x-3\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:

1. \begin{align*}2x^2+9x-18=0\end{align*}
2. \begin{align*}3x^2+8x-3=0\end{align*}
3. \begin{align*}-5x^2+13x+6=0\end{align*}
4. \begin{align*}2x^2-11x+5=0\end{align*}
5. \begin{align*}3x^2+8x-3=0\end{align*}
6. \begin{align*}x^2-x-20=0\end{align*}
7. \begin{align*}2x^2-7x+5=0\end{align*}
8. \begin{align*}3x^2+7x=-2\end{align*}
9. \begin{align*}2x^2-15=-x\end{align*}
10. \begin{align*}3x^2-10x=8\end{align*}
11. How could you use the graphs of a system of equations to solve \begin{align*}3x^2-10x=8\end{align*}?
12. What's the difference between a quadratic equation and a quadratic function?
13. Will a quadratic equation always have 2 solutions? Explain.
14. 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?
15. When does it make sense to use the graphing method for solving a quadratic equation?

### Vocabulary Language: English

Double Root

Double Root

A solution that is repeated twice.
Parabola

Parabola

A parabola is the characteristic shape of a quadratic function graph, resembling a "U".

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

A quadratic function is a function that can be written in the form $f(x)=ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.
Roots

Roots

The roots of a function are the values of x that make y equal to zero.
Zero Product Rule

Zero Product Rule

The zero product rule states that if the product of two expressions is equal to zero, then at least one of the original expressions much be equal to zero.
Zeroes

Zeroes

The zeroes of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.
Zeros

Zeros

The zeros of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.