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Why do quadratic equations where the discriminant is 0 have exactly 1 real solution?
Suppose the quadratic equation is
ax^2 + bx + c = 0 and D = b^2 - 4ac is the discriminant.
Then the solutions to the quadratic equation are [-b ± sqrt(d)]/(2a).
Since D = 0, the both solutions are equal to -b/(2a), a single real solution.
What else can I help you with?
Why a system of linear equations cannot have exactly two solutions?
A system of linear equations can only have: no solution, one solution, or infinitely many solutions.
How many times will the graph of a quadratic function cross or touch the x-axis if the discriminant is zero?
It will touch it at exactly 1 point. If a quadratic function is given as f(x) = ax2 + bx + c, let the discriminant be denoted as D. Then the graph of y = f(x) will cross the x-axis at the x-values x = (-b + sqrt(D))/(2a) and x = (-b - sqrt(D))/(2a). When the discriminant D = 0, these 2 x-values are actually the same. Thus the graph will touch the x-axis only once.
Why is it necessary to check for extraneous solutions in radical equations?
1) When solving radical equations, it is often convenient to square both sides of the equation. 2) When doing this, extraneous solutions may be introduced - the new equation may have solutions that are not solutions of the original equation. Here is a simple example (without radicals): The equation x = 5 has exactly one solution (if you replace x with 5, the equation is true, for other values, it isn't). If you square both sides, you get: x2 = 25 which also has the solution x = 5. However, it also has the extraneous solution x = -5, which is not a solution to the original equation.
A quadratic polynomial has a degree greater than 2?
true :) apex! * * * * * APEX gets it wrong - again! A quadratic polynomial has degree 2. Not greater than, nor less than but exactly equal to 2.
Use the substitution method to solve the system of equations. Choose the correct ordered pair. 2y plus 2x 20 y - 2x 4?
It is not possible to know exactly what the question is because the browser used by this site is almost totally useless for mathematical questions: it rejects most symbols. If the equations are 2y + 2x = 20 and y - 2x = 4,then the solution is (2, 8).
If the discriminant of an equation is zero then?
The term "discriminant" is usually used for quadratic equations. If the discriminant is zero, then the equation has exactly one solution.
Does a quadratic equation always have two solutions?
A quadratic equation can have two solutions, one solution, or no real solutions, depending on its discriminant (the part of the quadratic formula under the square root). If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution (a repeated root); and if it is negative, there are no real solutions, only complex ones. Thus, a quadratic equation does not always have two solutions.
If the discriminant equals zero the equation has solution(s).?
If the discriminant of a quadratic equation equals zero, it indicates that the equation has exactly one real solution, also known as a repeated or double root. This occurs because the quadratic touches the x-axis at a single point, rather than crossing it. Mathematically, this means that the two roots are the same, resulting in one unique solution for the equation.
If the discriminant of a quadratic equation equals zero is true of the equation?
If the discriminant of a quadratic equation equals zero, it indicates that the equation has exactly one real solution, also known as a repeated or double root. This means that the parabola represented by the quadratic equation touches the x-axis at a single point rather than crossing it. In this case, the quadratic can be expressed in the form ((x - r)^2 = 0), where (r) is the root.
Why are there usually two solutions to a quadratic equation?
In the graph of a quadratic equation, the plotted points form a parabola. This parabola usually intersects the X axis at two different points. Those two points are also the two solutions for the quadratic equation. Alternatively: Quadratic equations are formed by multiplying two linear equations together. Each of the linear equations has one solution - multiplying two together means that the solution for either is also a solution for the quadratic equation - hence you get two possible solutions for the quadratic unless both linear equations have exactly the same solution. Example: Two linear equations : x - a = 0 x - b = 0 Multiplied together: (x - a) ( x - b ) = 0 Either a or b is a solution to this quadratic equation. Hence most often you have two solutions but never more than two and always at least one solution.
If the discriminant of an equation is zero then what is true about the equation?
If the discriminant of a quadratic equation is zero, it indicates that the equation has exactly one real solution, also known as a double root. This means the parabola represented by the quadratic touches the x-axis at a single point rather than crossing it. In other words, the vertex of the parabola lies on the x-axis.
A system of equations with exactly one solution?
A system of equations with exactly one solution intersects at a singular point, and none of the equations in the system (if lines) are parallel.
How do you determine the nature of solutions?
To determine the nature of solutions for a mathematical equation, such as a quadratic equation, you can use the discriminant (D), which is calculated as (D = b^2 - 4ac). If (D > 0), there are two distinct real solutions; if (D = 0), there is exactly one real solution (a repeated root); and if (D < 0), there are no real solutions, but two complex solutions. This method can be applied to various types of equations to assess their solution types.
In the Quadratic Formula the expression under the radical sign b² 4ac is the .?
In the Quadratic Formula, the expression under the radical sign ( b^2 - 4ac ) is called the discriminant. It determines the nature of the roots of the quadratic equation ( ax^2 + bx + c = 0 ). If the discriminant is positive, there are two distinct real roots; if it is zero, there is exactly one real root; and if it is negative, the equation has two complex roots.
A system of two linear equations has exactly one solution if?
The slopes (gradients) of the two equations are different.
Why does a quadratic formula have two solutions?
A quadratic equation, typically in the form ( ax^2 + bx + c = 0 ), is a polynomial of degree two, which means its graph is a parabola. According to the Fundamental Theorem of Algebra, a polynomial of degree ( n ) has exactly ( n ) roots (solutions) in the complex number system. Therefore, a quadratic equation has two solutions, which can be real or complex, depending on the discriminant (( b^2 - 4ac )). If the discriminant is positive, there are two distinct real solutions; if it is zero, there is one real solution (a double root); and if it is negative, there are two complex solutions.
Why a system of linear equations cannot have exactly two solutions?
A system of linear equations can only have: no solution, one solution, or infinitely many solutions.
