Math

Discriminant and Nature of Roots In Quadratic Equation

A quadratic equation is an equation of the form ax2 + 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 ax2 + bx + c = 0 in terms of the coefficients of a, b, and c. Then the roots are now expressed as quadratic formula From the formula, the discriminant is b2 – 4ac. The value of b2 – 4ac determines the nature of the roots of the equation:
  • If b2 – 4ac is positive or greater than zero, the two roots are real and different;
  • If b2 – 4ac is zero, the two roots are real, and both equal to -b/2a
  • If b2 – 4ac is negative or less than zero, the quadratic equation has no real roots, i.e. the roots are complex numbers or imaginary.
From the conditions above, to have equal roots, it means that b2 – 4ac = 0. Read: Variation in mathematics and types

Using the roots

If α and β are the roots of the quadratic equation ax2 + 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 X2 + (b/a)x + c/a = 0 X2 – (α + β)x + αβ = 0 Examples (1) If the equation y2 + 3y + k = 0 has equal roots, the value of k is? As I have mentioned above, equal roots means b2 – 4ac = 0, a = 1, b = 3 , and c = k 32 – 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 y2 – (α + β)y + αβ = 0 The equation will be y2 – 5y -14 = 0

Bolarinwa Olajire

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