Write the formula for the discriminant.
State the types of roots for a quadratic equation, explaining how the discriminant helps you determine the type.
D = B^2 - 4AC
1. If B^2 is less than 4AC(B^2 < 4AC), the Determinant is negative and we have 2 imaginary solutions and no real solutions.
2. If B^2 = 4AC, we have one real solution and no imaginary solutions.
3.If B^2 > 4AC, we have 2 real solutions
and no imaginary solutions.
The formula for the discriminant of a quadratic equation is:
Δ = b² - 4ac
where Δ represents the discriminant, a, b, and c represent the coefficients of the quadratic equation.
Types of Roots for a Quadratic Equation:
1. If the discriminant (Δ) is greater than zero (Δ > 0), then the quadratic equation will have two distinct real roots. This means the equation will intersect the x-axis at two different points.
2. If the discriminant is equal to zero (Δ = 0), then the quadratic equation will have one real root. This means the equation will intersect the x-axis at a single point, resulting in a perfect square trinomial.
3. If the discriminant is less than zero (Δ < 0), then the quadratic equation will have no real roots. This means the equation will not intersect the x-axis, and the solutions of the quadratic equation will be complex numbers (imaginary roots). In this case, the quadratic equation will contain terms involving the imaginary unit, 'i'.
By using the discriminant, you can determine the type of roots a quadratic equation will have.
The formula for the discriminant of a quadratic equation is:
Δ = b^2 - 4ac
Where Δ represents the discriminant, b is the coefficient of the linear term, a is the coefficient of the quadratic term, and c is the constant term in the quadratic equation (ax^2 + bx + c = 0).
The discriminant helps determine the types of roots (or solutions) of a quadratic equation. Here are the three possibilities:
1. If the discriminant is positive (Δ > 0), then the equation has two distinct real roots. This means that when you solve the equation, you will find two different values for x.
2. If the discriminant is zero (Δ = 0), then the equation has one real root. In other words, there is only one value for x that satisfies the equation.
3. If the discriminant is negative (Δ < 0), then the equation has no real roots. Instead, the roots are complex conjugates. Complex roots involve imaginary numbers and are of the form a + bi and a - bi, where a and b are real numbers and i is the imaginary unit (√-1).
In summary, the discriminant allows us to determine the nature of the solutions to a quadratic equation.