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QUADRATIC EQUATIONS - Coggle Diagram
QUADRATIC EQUATIONS
Definition:
:star: In algebra, a quadratic equation (from the Latin quadratus for "square") is an equation that can be rearranged in standard form.
:star: Where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no. term.
The Nature of the Roots by Using Quadratic Formula:
:star: The quadratic formula is a formula that we can use to find the roots of the quadratic equation ax2 + bx + c = 0.
:star: To use the quadratic formula to find the roots of a quadratic equation, all we have to do is get our quadratic equation into the form ax2 + bx + c = 0; identify a, b, and c; and then plug them in to the formula.
:star: To identify these values, we just remember that a is in front of x2, b is in front of x, and c is the number by itself.
The Concept of Discriminant:
:star: If the discriminant is positive, then there are two distinct roots
:star: If the discriminant is zero, then there is exactly one real root
:star: If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots.
:star: In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta.
Conjugate of a Complex Number:
:star: The two complex numbers a+bi and a-bi are called conjugates of each other and the conjugate of z=a+bi is denoted by 𝑧̅=a-bi
NOTE:
:check: If a quadratic equation with real coefficients has a negative discriminant (∆< 0), the two solutions of the equation are complex conjugates of each other. If one of the roots of a quadratic equation is a+bi then the other one is a-bi.
The Relation between the Roots and the Coefficients of a Quadratic Equation:
:check: Let us take the quadratic equation of the general form ax^2 + bx + c = 0 where a (≠ 0) is the coefficient of x^2, b the coefficient of x and c, the constant term.
:check:Let α and β be the roots of the equation ax^2 + bx + c = 0
:check:Now we are going to find the relations of α and β with a, b and c.
:check: Now ax^2 + bx + c = 0
:check: Multiplication both sides by 4a (a ≠ 0) we get
:check: 4a^2x^2 + 4abx + 4ac = 0
:check: (2ax)^2 + 2
2ax
b + b^2 – b^2 + 4ac = 0
:check: (2ax + b)^2 = b^2 - 4ac
Forming a Quadratic Equation when the Roots are given:
:star: If α and β are the two roots of a quadratic equation, then the formula to construct the quadratic equation is
x2 - (α + β)x + αβ = 0
:star: That is,
x2 - (sum of roots)x + product of roots = 0
:star: If a quadratic equation is given in the standard form, we can find the sum and product of the roots using the coefficient of x2, x, and constant term.
Note :
:check: Irrational roots of a quadratic equation occur in conjugate pairs.
:check: That is, if (m + √n) is a root, then (m - √n) is the other root of the same quadratic equation.
