Q.If one root of $k(x-1)^2 = 5x-7$ is double the other root, show that $k=2$ or $-25$. OR Express the matrix $A = \begin{bmatrix}1 & 3 & 5\\-6 & 8 & 3\\-4 & 6 & 5\end{bmatrix}$ as the sum of a symmetric and a skew symmetric matrices.

For $P(x)=ax^2+bx+c$ ($a\ne0$), completing the square gives the quadratic formula:

Discriminant $D=b^2-4ac$: $D>0$ gives two distinct real roots (parabola crosses the x-axis twice); $D=0$ gives one repeated real root (touches once); $D<0$ gives no real roots (a complex conjugate pair; parabola never meets the x-axis).

Sum and product of roots. For roots $\alpha,\beta$ of $ax^2+bx+c=0$: $\alpha+\beta=-\dfrac ba$, $\alpha\beta=\dfrac ca$. This pair of formulas is the single most useful tool in the chapter for:

• Building an equation from its roots: $x^2-(\alpha+\beta)x+\alpha\beta=0$;
• Transforming roots (reciprocals $1/\alpha,1/\beta$; doubles $2\alpha,2\beta$; negatives $-\alpha,-\beta$) into a new equation without ever solving for $\alpha,\beta$ explicitly; …