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The discriminant, d, is the name given to the expression \( b^{2} - 4ac \), which appears under the radical sign in the quadratic formula. Since \( d = b^{2} - 4ac \), you can rewrite the quadratic formula as \( x = \frac{- b \pm \sqrt{d}}{2a} \).
You can use the discriminant to determine the type and number of roots of a quadratic equation.
| Discriminant Value | Number and Type of Roots |
|---|---|
| Positive \( (d > 0) \) | Two real number roots |
| Zero \( (d = 0) \) | One real number root |
| Negative \( (d < 0) \) | Two complex number roots |
Use the activity below to see how well you can find the discriminant of a quadratic equation. On each tab, find the discriminant and then state the number and type of roots. Then give the root(s). Be sure to check your answer.
Use the discriminant to find the number and type of roots of \( x^{2} + 6x + 9 = 0 \). Then give the roots.
The discriminant is 0, so there is one real number root. It is \( x = - 3 \).
If you need help arriving at this answer, click the solution button.
This quadratic equation is written in standard form. Identify the values of a, b, and c. |
\( a = 1 \) \( b = 6 \) \( c = 9 \) |
Substitute the values into the discriminant. |
\( b^{2} - 4ac \) \( {(6)}^{2} - 4\left( 1 \right)\left( 9 \right) = \) \( 36 - 36 = 0 \) Since \( d = 0, \) there is one real number solution. |
Use \( x = \frac{- b \pm \sqrt{d}}{2a} \) to find the value of the root. |
\( x = \frac{- b \pm \sqrt{d}}{2a} \) \( x = \frac{- (6) \pm \sqrt{0}}{2(1)} \) \( x = \frac{- 6}{2} = - 3 \) |
Use the discriminant to find the number and type of roots of \( {3x}^{2} - 2x + 4 = 0 \). Then give the roots.
The discriminant is negative, so there are two complex number roots. They are \( x = \frac{1 \pm i\sqrt{11}}{3} \).
If you need help arriving at this answer, click the solution button.
This quadratic equation is written in standard form. Identify the values of a, b, and c. |
\( a = 3 \) \( b = - 2 \) \( c = 4 \) |
Substitute the values into the discriminant. |
\( b^{2} - 4ac \) \( \left( - 2 \right)^{2} - 4\left( 3 \right)\left( 4 \right) = \) \( 4 - 48 = - 44 \) Since \( d < 0, \) there are two complex number solutions. |
Use \( x = \frac{- b \pm \sqrt{d}}{2a} \) to find the value of the root. |
\( x = \frac{- b \pm \sqrt{d}}{2a} \) \( x = \frac{- ( - 2) \pm \sqrt{- 44}}{2(3)} \) \( x = \frac{2 \pm \sqrt{- 1}\sqrt{4}\sqrt{11}}{6} \) \( x = \frac{2 \pm 2i\sqrt{11}}{6} \) \( x = \frac{1 \pm i\sqrt{11}}{3} \) |
