Math

# Discriminant and Nature of Roots In Quadratic Equation

A quadratic equation is an equation of the form ax

If α and β are the roots of the quadratic equation ax

^{2}+ bx + c = 0. The values of x that satisfy the quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by factorization, completing the square, or using the formula. Before I can explain the discriminant, there is a need to first express the quadratic formula. The formula makes it possible to express the roots of a general quadratic equation of the form ax^{2}+ bx + c = 0 in terms of the coefficients of a, b, and c. Then the roots are now expressed as From the formula, the discriminant is b^{2}– 4ac. The value of b^{2}– 4ac determines the nature of the roots of the equation:- If b
^{2}– 4ac is positive or greater than zero, the two roots are real and different; - If b
^{2}– 4ac is zero, the two roots are real, and both equal to -b/2a - If b
^{2}– 4ac is negative or less than zero, the quadratic equation has no real roots, i.e. the roots are complex numbers or imaginary.

^{2}– 4ac = 0. Read: Variation in mathematics and types**Using the roots**

If α and β are the roots of the quadratic equation ax^{2}+ bx + c = 0, it can be shown that α + β = -b/a and αβ = c/a Once the roots α and β are known, the quadratic equation can then be written as X^{2}+ (b/a)x + c/a = 0 X^{2}– (α + β)x + αβ = 0 Examples (1) If the equation y^{2}+ 3y + k = 0 has equal roots, the value of k is? As I have mentioned above, equal roots means b^{2}– 4ac = 0, a = 1, b = 3 , and c = k 3^{2}– 4(1*k) = 0 9 – 4k = 0 4k = 9 K = 9/4 (2) Find the quadratic equation whose roots are y = 7 and y = -2 α + β = 7 + (-2) = 7 – 2 = 5 αβ = 7 * (-2) = -14 Using y^{2}– (α + β)y + αβ = 0 The equation will be y^{2}– 5y -14 = 0