x |
f(x) |
0 |
6 |
1 |
0 |
4 |
-6 |
6 |
0 |
7 |
6 |
The solutions to quadratic equations are called roots. Roots are the x-intercepts or zeros of a quadratic function.
For every quadratic equation there is a related quadratic function. For example, if you are given the quadratic equation x2 + 5x + 4 = 0, the related quadratic function is f(x) = x2 + 5x + 4.
A quadratic equation may have two solutions, one solution, or no solution. In this section, you will only learn about quadratics with one or two solutions.
Make a table of values for the function f(x) = x2 – 7x + 6.
x |
f(x) |
0 |
|
1 |
|
4 |
|
6 |
|
7 |
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x |
f(x) |
0 |
6 |
1 |
0 |
4 |
-6 |
6 |
0 |
7 |
6 |
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f(x) = 0
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x = 1 and x = 6
Click on the table to see how the value for x, that you have identified, generates the solutions to the equation x2 – 7x + 6 = 0.
How can you identify the solutions to a quadratic equation from a table if the quadratic equation consists of a polynomial that is equal to 0?
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Look for values of the variable (usually x) that generate the function value of 0.
If the quadratic equation consists of a polynomial that is equal to b instead of 0, how would you use a table of values to identify the solutions to that equation?
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Look for values of the variable (usually x) that generate the function value of b.
Use the tables to determine the zero of each function below. The zero of a function is the root, or solution, to the related equation when the function is equal to 0.
t |
f(t) |
-4 |
8 |
-3 |
0 |
3 |
-6 |
4 |
0 |
5 |
8 |
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The related equation would be the quadratic function when it is set to be equal to 0. Look for function values that are equal to 0.
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The roots of the quadratic function, f(t) = t2 – t – 12, are t = 4 and t = −3, which are the t-values of the ordered pairs (4, 0) and (−3, 0) from the table.
0 = 42 – 4 – 12
0 = 16 – 4 – 12
0 = 0
0 = (−3)2 – (−3) – 12
0 = 9 + 3 – 12
0 = 0
x |
f(x) |
-3 |
64 |
-1 |
36 |
0 |
25 |
4 |
1 |
5 |
0 |
6 |
1 |
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The related equation would be the quadratic function when it is set to be equal to 0. Look for function values that are equal to 0.
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It appears from the table that the quadratic function, f(x) = x2 – 10x + 25, has only one zero, x = 5, which is the x-value of the ordered pair (5, 0) from the table.
0 = 52 – 10(5) + 25
0 = 25 – 50 + 25
0 = 0
x |
f(x) |
-2 |
3 |
-1 |
0 |
0 |
-1 |
1 |
0 |
2 |
3 |
3 |
8 |
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The related equation would be the quadratic function when it is set to be equal to 0. Look for function values that are equal to 0.
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The solutions of the quadratic equation, x2 – 1 = 0, are x = −1 and x = 1, which are the x-values of the ordered pairs (−1, 0) and (1, 0) from the table.
0 = (-1)2 – 1
0 = 1 – 1
0 = 0
0 = 12 – 1
0 = 1 – 1
0 = 0