Final answer:
To evaluate the discriminant for a quadratic equation with the given values a = 1, b = -7, and c = 4, we calculate (-7)^2 - 4(1)(4), which equals 33. This positive discriminant value indicates that the quadratic equation has two distinct real roots.
Step-by-step explanation:
The question involves evaluating the discriminant of a quadratic equation, which is given by the expression b^2 - 4ac. Given the values a = 1, b = -7, and c = 4, we substitute these into the formula to get the discriminant value:
(-7)^2 - 4(1)(4) = 49 - 16 = 33.
The discriminant value is 33. This number is important as it tells us the nature of the roots of the quadratic equation. Since 33 is a positive number, it indicates that the equation has two distinct real roots.
The discriminant is a value derived from the coefficients of a quadratic equation and provides information about the nature of its roots (solutions). For a quadratic equation in the form
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The discriminant can take on three different scenarios:
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D>0: In this case, the quadratic equation has two distinct real roots. This happens when the graph of the quadratic equation intersects the x-axis at two different points.
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D=0: The equation has one real root (also known as a repeated or double root). The graph of the quadratic equation touches the x-axis at a single point.
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D<0: The equation has no real roots. In this scenario, the graph of the quadratic equation does not intersect the x-axis, and the solutions are complex numbers.