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The Discriminant and the Quadratic Graph The discriminant of the quadratic equation $ax^2+bx+c=0$ is $\Delta = b^2-4ac$. We used the discriminant to determine the number of real roots of the quadratic equation. If they exist, these roots correspond to $x$-intercepts of the quadratic $y = ax^2+bx+c$. The discriminant tells us about the relationship between a […]
The formula $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$ gives the solutions to the general quadratic equation $ax^2+bx+c=0$. By examining the expression under the square root sign, $b^2-4ac$, we known as the discriminant, symbol used is $\Delta$. The quadratic formula becomes $x=\dfrac{-b + \sqrt{\Delta}}{2a}$ and $x=\dfrac{-b – \sqrt{\Delta}}{2a}$. $\Delta <0$ If $x^2+2x+3=0$, then $a=1,b=2$ and $c=3$. \( […]
Discriminant can be used to ellipses for identifying the status of their intersections in conjunction with eccentricity. The eccentricity of \( \displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is \( \displaystyle e^2 = 1 – \frac{b^2}{a^2} \). Worked Example of Ellipse Discriminant Eccentricity (a) The tangent \( \ell \) has equation \( y=mx+k\). Show that […]