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QUADRATIC EQUATIONS - Coggle Diagram
QUADRATIC EQUATIONS
Definition of Quadratic Equation
A
quadratic equation
is an equation of the second degree, meaning it contains at least one term that is squared. The standard form is ax² + bx + c = 0 with a, b, and c being constants, or numerical coefficients, and x is an unknown variable. One absolute rule is that the first constant "a" cannot be a zero.
The Nature of the Roots by Using Quadratic Formula
The discriminant determines the nature of the roots of a quadratic equation. The word 'nature' refers to the types of numbers the roots can be: namely real, rational, irrational or imaginary. If Δ≥0, the expression under the square root is non-negative and therefore roots are real.
The Concept of Discriminant
The discriminant is the term underneath the square root in the quadratic formula and tells us the number of solutions to a quadratic equation. If the discriminant is positive, we know that we have 2 solutions. If it is negative, there are no solutions and if the discriminant is equal to zero, we have one solution.
Expression of a Complex Number
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a+bi a + b i where a is the real part and bi is the imaginary part. For example, 5+2i 5 + 2 i is a complex number.
Imaginary Number
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1.
Set of Complex Numbers
The set of complex numbers, denoted by C, includes the set of real numbers (R) and the set of pure imaginary numbers.
Conjugate of a Complex Number
You find the complex conjugate simply by changing the sign of the imaginary part of the complex number. To find the complex conjugate of 4+7i we change the sign of the imaginary part. Thus the complex conjugate of 4+7i is 4 - 7i. To find the complex conjugate of 1-3i we change the sign of the imaginary part.
The Relation between the Roots and the Coefficients of a Quadratic Equation
The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term, divided by the leading coefficient. The product of the roots of a quadratic equation is equal to the constant term (the third term), divided by the leading coefficient.
Forming a Quadratic Equation when the Roots are given
To form a quadratic equation, let α and β be the two roots. Let us assume that the required equation be ax2 + bx + c = 0 (a ≠ 0).
