The quadratic formula

Functions: Quadratic functions

The quadratic formula

By completing the square we can rewrite the quadratic expression $ax^2+bx+c$ as $a\cdot\left(\left(x+\dfrac{b}{2a}\right)^2-\dfrac{b^2-4a\cdot c}{(2a)^2}\right)\tiny.$ We use this formula to solve quadratic equations.

Discriminant and quadratic formula

Let $a$ , $b$ , and $c$ be real numbers with $a\ne0$ . The equation $ax^2+bx+c=0$ can be reduced to

$x=\dfrac{-b - \sqrt{b^2-4a\cdot c}}{2a}\lor x=\dfrac{-b+ \sqrt{b^2-4a\cdot c}}{2a}$
The expression $b^2-4ac$ is called the discriminant of $ax^2+bx+c$ , and is often referred to with $D$ .

With this notation $x=\dfrac{-b - \sqrt{D}}{2a}\lor x=\dfrac{-b+ \sqrt{D}}{2a}$

If $D\lt 0$ , then there are no solutions.
If $D= 0$ , then there is only one solution: $x=\dfrac{-b}{2a}$ .
If $D\gt 0$ , then there are two solutions.

The above is also called the quadratic formula. With this formula any quadratic equation can be solved.

The reduction is as follows:

$\begin{array}{rcl}ax^2+bx+c&=&0\\ &&\phantom{xx}\color{blue}{\text{the original equation}}\\ a\cdot\left(\left(x+\dfrac{b}{2a}\right)^2-\dfrac{D}{(2a)^2}\right)&=&0\\ &&\phantom{xx}\color{blue}{\text{the square completed, with }D = b^2-4a\cdot c}\\ \left(x+\dfrac{b}{2a}\right)^2-\dfrac{D}{(2a)^2}&=&0\\ &&\phantom{xx}\color{blue}{\text{left and right hand side divided by }a\text{ }}\\ \left(x+\dfrac{b}{2a}\right)^2&=&\dfrac{D}{(2a)^2}\\ &&\phantom{xx}\color{blue}{\text{constant term to the right hand side }}\\ x+\dfrac{b}{2a}&=&\pm\sqrt{\dfrac{D}{(2a)^2}}\\ &&\phantom{xx}\color{blue}{\text{root taken; }\pm\text{means: }+\text{ of }-}\\ x+\dfrac{b}{2a}&=&\pm\dfrac{\sqrt{D}}{2a}\\ &&\phantom{xx}\color{blue}{\text{root simplified}}\\ x&=&-\dfrac{b}{2a}\pm\dfrac{\sqrt{D}}{2a}\\ &&\phantom{xx}\color{blue}{\text{constant term to the right hand side}}\\ \end{array}$

The quadratic formula can be used to determine the intersection points of a parabola with the $x$ -axis. What does this mean for the three cases which we have just seen? We will still consider a parabola opening upwards.

If $D\gt0$ , then the square root is taken from a positive number, so the equation $ax^2+bx+c=0$ has two different solutions. The parabola, that is, the graph of $ax^2+bx+c$ , intersects the $x$ -axis twice.

If $D\lt0$ , then there are no solutions because taking the square root of a negative number does not have any real numbers as solution. The parabola will not intersect the $x$ -axis.

If $D= 0$ , then both solutions are equal and there is only one solution: $x=\dfrac{-b}{2a}$ . Here the parabola touches the $x$ -axis.

Solve for $x$ : $3 x^2-6 x-72=0$

Give your answer in the form $x=a\lor x=b$ .

$x=-4\lor x=6$

Use the quadratic formula:
$x=\frac{-b-\sqrt{D}}{2a}\lor x=\frac{-b+\sqrt{D}}{2a}$
with discriminant $D=b^2-4a\cdot c$ .

$\begin{array}{rcl} a=3&b=-6&c=-72\\ &&\phantom{xx}\color{blue}{\text{a, b and c determined}}\\ (-6)^2-4\cdot 3\cdot -72&=&900\\ &&\phantom{xx}\color{blue}{\text{discriminant calculated}}\\ x=\frac{6-30}{6}&\lor&x=\frac{6+30}{6}\\ &&\phantom{xx}\color{blue}{\text{quadratic formula used}}\\ x=-4&\lor&x=6\\ &&\phantom{xx}\color{blue}{\text{solution simplified}}\\ \end{array}$ The graph of the function $y=3 x^2-6 x-72$ is drawn below. The $x$ -coordinates of the meeting points of the graph and the $x$ -axis are the solutions to the equation.

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