3FG1MoreExamples

Factorising and Solving Complex Equations

More Examples


 * Example 1 **

Factorise over C by completing the square math . \qquad z^2+6z+16 math


 * Solution:**


 * Complete the square on the first two terms by adding 9,
 * then subtract 9 again to keep the equation balanced.

math \\ . \qquad z^2+6z+16 \\. \\ . \qquad =\underline{z^2+6z+9}+16-9 \\. \\ . \qquad =\left( z+3 \right)^2 + 7 math

Introduce i 2 = –1 to form a difference of two squares.

math \\ . \qquad =\left( z+3 \right)^2 - 7 \, i^2 \\. \\ . \qquad =\left( z+3+\sqrt{7} \, i \right) \left( z+3-\sqrt{7} \, i \right) math

Example 2

Factorise over C by completing the square math . \qquad 3z^2+z+12 math and hence solve the equation math . \qquad 3z^2+z+12=0 math


 * Solution:**


 * Take out 3 as a common factor,
 * then complete the square on the first two terms.

math \\ . \qquad 3z^2+z+12 \\. \\ . \qquad =3\left( z^2 + \frac{1}{3} z+4 \right) \\. \\ . \qquad =3\left( \underline{z^2 + \frac{1}{3}z+\frac{1}{36}}+4-\frac{1}{36} \right) \\. math math . \qquad =3\left[ \left( z+\dfrac{1}{6} \right)^2 + \dfrac{143}{36} \right] math

Introduce i 2 = –1 to form a difference of two squares

math . \qquad =3\left[ \left( z+\dfrac{1}{6} \right)^2 - \dfrac{143 \, i^2}{36} \right] \\. math math . \qquad =3 \left( z+ \dfrac{1}{6} + \dfrac{\sqrt{143} \, i}{6} \right) \left( z+ \dfrac{1}{6} - \dfrac{\sqrt{143} \, i}{6} \right) math

Hence solve the equation math . \qquad 3z^2+z+12=0 \\. math math . \qquad 3 \left( z+ \dfrac{1}{6} + \dfrac{\sqrt{143} \, i}{6} \right) \left( z+ \dfrac{1}{6} - \dfrac{\sqrt{143} \, i}{6} \right)=0 \\. math math . \qquad z = -\dfrac{1}{6} \pm \dfrac{\sqrt{143} \, i}{6} math





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