Question Video: Solving Nonmonic Quadratic Equations with Complex Roots Mathematics • 10th Grade

Video Transcript

Solve the quadratic equation four 𝑥 squared plus three 𝑥 plus one is equal to zero.

Our equation is written in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero. One way of solving this is using the quadratic formula, which states that 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎. In this question, 𝑎 is equal to four, 𝑏 is equal to three, and 𝑐 is equal to one. They are the coefficients of 𝑥 squared, coefficient of 𝑥, and constant term, respectively. Substituting in our values, we have 𝑥 is equal to negative three plus or minus the square root of three squared minus four multiplied by four multiplied by one all divided by two multiplied by four.

Three squared is equal to nine. Four multiplied by four multiplied by one is 16. And two multiplied by four is equal to eight. This can be simplified further so that 𝑥 is equal to negative three plus or minus the square root of negative seven all divided by eight. We know that the square root of negative seven can be rewritten as the square root of seven multiplied by the square root of negative one. From our knowledge of complex numbers, the square root of negative one is equal to 𝑖.

This means that the square root of negative seven is equal to the square root of seven 𝑖. This means that we have two possible solutions for 𝑥. Either 𝑥 equals negative three plus root seven 𝑖 over eight or 𝑥 is equal to negative three minus root seven 𝑖 over eight. These are the two solutions to the quadratic equation four 𝑥 squared plus three 𝑥 plus one is equal to zero.