This information is useful as it serves as a double check when we solve quadratic equations by any of the four techniques. Question 4: Why is the discriminant important?Īnswer: The quadratic equation discriminant is significant since it tells us the number and kind of solutions. Moreover, quadratics of either type do not ever take the value 0, thus their discriminant is negative. Question 3: What is a negative quadratic?Īnswer: A quadratic expression that always takes positive values is referred to as positive definite, while one that always takes negative values is referred to as negative definite. A negative discriminant denotes that neither of the solutions is real numbers. A discriminant of zero denotes that the quadratic consists of a repeated real number solution. Question 2: What does a negative discriminant mean?Īnswer: A positive discriminant denotes that the quadratic has two different real number solutions. If the discriminate is negative, the roots will be imaginary. The roots may be imaginary, real, unequal or equal.
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Question 1: What are the nature of roots?Īnswer: The nature of roots simply means the category in which the roots are falling upon. Soution: B) The Discriminant of the given equation = 0 What are three different methods to solve Quadratic Equations? More Solved Examples For YouĮxample 3: Determine the value(s) of p for which the quadratic equation 2 x 2 + p x + 8 = 0 has equal roots:Ī) p = ☖4 B) p = ☘ C) p = ±4 D) p = ☑6 Hence, here we have understood the nature of roots very clearly. Solution: The discriminant D of the given equation isĬlearly, the discriminant of the given quadratic equation is zero. Therefore, the roots of the given quadratic equation are real, irrational and unequal.Įxample 2: Without solving, examine the nature of roots of the equation 4x 2 – 4x + 1 = 0?
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The discriminant D of the given equation isĬlearly, the discriminant of the given quadratic equation is positive but not a perfect square. Solution: Here the coefficients are all rational. Let us just summarize all the above cases in this table below: b 2 – 4ac > 0ī 2 – 4ac > 0 (is aperfect square and a or b is irrational)Įxample 1: Discuss the nature of the roots of the quadratic equation 2x 2 – 8x + 3 = 0. When a, b, and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax 2 + bx + c = 0 are irrational. Case VI: b 2 – 4ac > 0 is perfect square and a or b is irrational.Here the roots α and β form a pair of irrational conjugates. When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive but not a perfect square then the roots of the quadratic equation ax 2 + bx + c = 0 are real, irrational and unequal. Case V: b 2 – 4ac > 0 and not perfect square.When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax 2 + bx + c = 0 are real, rational and unequal. Case III: b 2– 4ac 0 and perfect square.When a, b, and c are real numbers, a ≠ 0 and the discriminant is zero, then the roots α and β of the quadratic equation ax 2+ bx + c = 0 are real and equal. When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive, then the roots α and β of the quadratic equation ax 2 +bx+ c = 0 are real and unequal. Let us recall the general solution, α = (-b-√b 2-4ac)/2a and β = (-b+√b 2-4ac)/2a
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Depending on the values of the discriminant, we shall see some cases about the nature of roots of different quadratic equations. Its value determines the nature of roots as we shall see. Hence, the expression (b 2 – 4ac) is called the discriminant of the quadratic equation ax 2 + bx + c = 0. Therefore for this equation, there are no real number solutions. There is no real number whose square is negative. We say this because the root of a negative number can’t be any real number.
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Hence, the nature of the roots α and β of equation ax 2 + bx + c = 0 depends on the quantity or expression (b 2 – 4ac) under the square root sign. Let α and β be the roots of the general form of the quadratic equation :ax 2 + bx + c = 0. The number of roots of a polynomial equation is equal to its degree.