The discriminant is the part of the quadratic formula
found within the square root. For a quadratic of the form aπ₯2 + bπ₯ + c, its discriminant is b2 β 4ac. A quadratic equation has 2, 1 or 0 solutions depending if the value of the discriminant is positive, zero or negative respectively. The discriminant, b2 β 4ac is represented by the delta symbol, Ξ. The discriminant formula is Ξ = b2 β 4ac, where a is the coefficient of π₯2, b is the coefficient of π₯ and c is the constant term of a quadratic. For example, calculate the discriminant of y = π₯2 + 5π₯ + 2.We have one π₯2. The coefficient of π₯2 is 1. Therefore a = 1. The coefficient of π₯ is 5. Therefore b = 5. The constant term is 2. Therefore c = 2. We substitute the values of a = 1, b = 5 and c = 2 into the formula for the discriminant, b2 β 4ac. b2 = 52 = 25 and 4ac = 4 Γ 1 Γ 2 = 8. b2 β 4ac becomes 28 β 8 = 17. The discriminant is 17. The discriminant is important because it tells us how many solutions any quadratic equation has.
The number of solutions to a quadratic equation tells us the number of roots of the quadratic equation. The roots of a quadratic equation are the locations where the quadratic graph crosses the π₯-axis. They are the π₯-axis intercepts. The following table shows the number of roots for a positive, negative or zero discriminant.
How to Calculate the DiscriminantTo calculate the discriminant of a quadratic equation, the formula is b2 β 4ac. Substitute the values of a, b and c after reading them from a quadratic equation of the form aπ₯2 + bπ₯ + c. For example, for π₯2 β 3π₯ + 4, a = 1, b = -3 and c = 4. b2 = 9 and 4ac = 16. The discriminant, b2 β 4ac = β 7. When calculating the discriminant it is important to consider these key points:
To find the discriminant:
Discriminant CalculatorTo use the discriminant calculator:
A Positive DiscriminantA positive discriminant means that the value of b2 β 4ac is greater than zero. A quadratic equation with a positive discriminant has exactly two solutions, which means that it has two π₯-axis intercepts. This means that the quadratic has 2 roots. A positive value of the discriminant means that the graph of the quadratic equation must pass through the π₯-axis twice. If the quadratic has a positive coefficient of π₯2 (a > 0), the graph is concave up and the minimum point will be below the π₯-axis as shown in the left image below. If the quadratic has a negative coefficient of π₯2 (a < 0), the graph is concave down and the minimum point will be above the π₯-axis as shown in the right image below. A positive value of the discriminant tells us that a quadratic has two unique solutions. For example, the quadratic π₯2 β 4π₯ + 3 = 0 has two solutions: π₯ = 3 and π₯ = 1. This means that the quadratic equation crosses the π₯-axis at π₯ = 1 and π₯ = 3. The discriminant value is 4, which is a positive number. The square root of a positive number has both a positive and negative answer. Therefore the quadratic formula provides 2 different solutions. If the discriminant is a perfect square, then the solutions to the quadratic equation are rational. This means that the solutions will be integers or can be written as fractions. If a discriminant is not a perfect square, the 2 solutions will be irrational. This is because the square root of this discriminant will be a surd. A discriminant of zero means that the value of b2 β 4ac is equal to zero. A quadratic equation with a discriminant of zero has exactly one solution. This means that the graph of the quadratic just touches the π₯-axis at its minimum or maximum point. A discriminant equal to zero means that the graph of the quadratic equation must touch the π₯-axis once. It cannot pass through the π₯-axis. Instead it just touches it at its one root. If the quadratic has a positive coefficient of π₯2 (a > 0), the graph is concave up and the minimum point will touch π₯-axis as shown in the left image below. If the quadratic has a negative coefficient of π₯2 (a < 0), the graph is concave down and the maximum point will touch the π₯-axis as shown in the right image below. For a quadratic equation with a discriminant of zero, there will be exactly one solution. This is because the square root of the discriminant is taken as part of the quadratic formula. The square root of 0 is 0. This means that we add or subtract 0 in the calculation, which causes the two answers to be the same. The solution to a quadratic equation with a discriminant of zero is called a repeated root. This is because the same solution appears twice. For example, the quadratic π₯2 β 4π₯ + 4 = 0 has a = 1, b = -4 and c = 4. b2 = 16 and 4ac = 16. b2 β 4ac = 0 and the one repeated root of the quadratic equation is π₯ = 2. A Negative DiscriminantA negative discriminant means that the value of b2 β 4ac is less than zero. When using the quadratic formula the square root of a negative cannot be found. There are no real solutions to the quadratic equation. The two solutions are complex and cannot be seen on a graph. A negative discriminant means that the graph of the quadratic equation does not touch the π₯-axis. If the quadratic has a positive coefficient of π₯2 (a > 0), the graph is concave up and the entirety of the graph is above the π₯-axis. All outputs of the graph are positive. If the quadratic has a negative coefficient of π₯2 (a < 0), the graph is concave down and the entirety of the graph is below the π₯-axis. All outputs of the graph are negative. For example, in the quadratic equation π₯2 β 3x + 5 =0, a = 1, b = -3 and c = 5. b2 = 9 and 4ac = 20. Therefore the discriminant b2 β 4ac = -11. The square root of a negative number cannot be found and so, no real solutions can be found since we cannot complete the calculation. The square root of -11 must be written in terms of the imaginary unit, i. Therefore it is written as β11 i. This complex solution can be used. The Discriminant from a GraphThe number of π₯-axis intercepts indicates the value of the discriminant:
The Discriminant of a Cubic EquationThe discriminant of a cubic equation aπ₯3 + bπ₯2 + cπ₯ + d is b2c2 β 4ac3 β 4b3d β 27a2d2 + 18abcd. If the discriminant is positive, the cubic has 3 real roots. If it is negative, the cubic has one real root and two complex conjugate roots. If it is zero, at least two of the roots are equal.
For example for the cubic 5π₯3 + 2π₯2 β 3π₯ + 1, a = 5, b = 2, c = -3 and d = 1. Using the cubic discriminant formula Ξ = b2c2 β 4ac3 β 4b3d β 27a2d2 + 18abcd. Ξ = -671. The cubic discriminant is negative and so, this cubic has 1 real root and 2 complex roots. This means that its graph will only intersect the π₯-axis once. What is the value of the discriminant of the quadratic equation 2/8 And what does its value mean about the number of real number solutions the equation has?What is the value of the discriminant of the quadratic equation -2x2=-8x+8 , and what does its value mean about the number of real number solutions the equation has? The discriminant is equal to 0, which means the equation has no real number solutions.
What is the value of the discriminant of the quadratic equation?For the quadratic equation ax2 + bx + c = 0, the expression b2 β 4ac is called the discriminant. The value of the discriminant shows how many roots f(x) has: - If b2 β 4ac > 0 then the quadratic function has two distinct real roots. - If b2 β 4ac = 0 then the quadratic function has one repeated real root.
What is the value of the discriminant for the quadratic equation =Summary: The value of the discriminant for the quadratic equation -3 = -x2 + 2x is 16.
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