Quadratic Equations: Real And Rational Roots With Perfect Square Discriminants

if the discriminant is > or = to 0 and a perfect square

rational

The discriminant is the part of the quadratic formula that appears under the square root sign. It is given by the expression b^2 – 4ac, where a, b, and c are the coefficients of the quadratic equation of the form ax^2 + bx + c = 0.

If the discriminant is ≥ 0 and a perfect square, then we can say that the quadratic equation has real and rational roots. This means that the graph of the quadratic equation will intersect the x-axis at two distinct points, and these points will be rational numbers.

If the discriminant is a perfect square, then we can write the expression b^2 – 4ac as (b – √d)(b + √d), where d is the perfect square. When we apply the quadratic formula, we get two roots x = (-b ± √d)/(2a), which can be simplified to x = (-b/2a) ± (√d/2a).

In this case, the discriminant is a perfect square, which means that √d is a rational number. Therefore, we get two rational roots for the quadratic equation, and we can write them as x = (-b ± k)/(2a), where k is a rational number.

For example, let’s consider the quadratic equation 3x^2 + 4x + 1 = 0. The discriminant in this case is b^2 – 4ac = 4^2 – 4(3)(1) = 4, which is a perfect square. Therefore, the quadratic equation has two real and rational roots, which we can find using the quadratic formula:

x = (-4 ± √4)/(2(3)) = (-4 ± 2)/6 = (-2/3) and (-1/3)

Therefore, the roots of the equation are x = -2/3 and x = -1/3.

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