QQuestionMathematics
QuestionMathematics
If the discriminant of a quadratic equation is 2, then the equation has _____.
If the discriminant of a quadratic equation is 4, then the equation has _____.
If the discriminant of a quadratic equation is 0, then the equation has _____.
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Answer
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Step 1:
The discriminant of a quadratic equation given by $ax^2 + bx + c = 0$ is calculated by the formula $b^2 - 4ac$.
When the discriminant is 2, it means that $b^2 - 4ac = 1$. Since the discriminant is positive, the quadratic equation has two distinct real roots. However, we cannot determine the exact values of the roots with just the discriminant value.
Step 2:
This is because the discriminant is a perfect square ($(2)^2 = 4$).
This indicates that the quadratic equation has two distinct real roots, and the roots are rational and opposite in sign (i.e., one root is the negative of the other).
Step 3:
When the discriminant of a quadratic equation is 0, it means that $b^2 - 4ac = 0$.
This has two implications: a. The quadratic equation has exactly one real root. b. The real root is a repeated root, i.e., the graph of the quadratic equation touches the x-axis at the root.
Final Answer
1. If the discriminant of a quadratic equation is 2, then the equation has two distinct real roots. 2. If the discriminant of a quadratic equation is 4, then the equation has two distinct rational and opposite roots. 3. If the discriminant of a quadratic equation is 0, then the equation has exactly one real repeated root.
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