Find two quadratic equations having the given solutions. (There are many correct answers.) 3+4 i

step 1 Here we have to find two quadratic equations which will have the solutions given by x is equals

Find two quadratic equations having the given solutions....

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Key Concepts

Formation of Quadratic Equations from Given Roots

Any quadratic equation can be constructed from its roots by writing it in the factored form (x - r1)(x - r2) = 0. This method emphasizes that the roots are exactly the numbers that nullify the equation, and multiplying out the factors gives the quadratic in standard form. Moreover, multiplying the equation by any nonzero constant yields another valid equation with the same roots.

Complex Conjugate Roots

When a quadratic equation has real coefficients, its non-real roots must appear in complex conjugate pairs. This means that if a + bi is a root, then its complex conjugate a - bi must also be a root. This attribute ensures that the imaginary parts cancel out when the equation is expanded, leaving a polynomial with real coefficients.

Vieta's Formulas

Vieta's formulas link the coefficients of a quadratic equation to the sum and product of its roots. In the standard form ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. These relationships offer a quick and powerful tool for forming or verifying quadratic equations when the roots are known.

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