For ax2+bx+c=0, a 0, the real roots are x= -b+-sqrt(b2-4ac)2a when b2-4ac 0. Completing the square gives (2ax+b)2=b2-4ac. A real square cannot be negative; a positive right side gives two square-root signs. Rearrange to zero; record signed a,b,c; substitute with brackets; simplify the square root; apply restrictions. Round only at the end if an approximation is requested. Example x2+2x-2=0 x= -2+-sqrt(4+8)2 =-1+- 3 Exam check: -b changes the sign of the entire coefficient b. A negative discriminant means no real roots, not a negative square root answer.

For \(ax^2+bx+c=0\), \(a\ne0\), the real roots are \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) when \(b^2-4ac\ge0\).

Completing the square gives \((2ax+b)^2=b^2-4ac\). A real square cannot be negative; a positive right side gives two square-root signs.

Rearrange to zero; record signed \(a,b,c\); substitute with brackets; simplify the square root; apply restrictions. Round only at the end if an approximation is requested.

\(\begin{aligned}x^2+2x-2&=0\\x&=\dfrac{-2\pm\sqrt{4+8}}2\\&=-1\pm\sqrt3\end{aligned}\)

Exam check: \(-b\) changes the sign of the entire coefficient \(b\). A negative discriminant means no real roots, not a negative square root answer.

Choosing and using the quadratic formula

Choosing and using the quadratic formula