## Definition

The quadratic formula gives us the solutions, or roots, to a quadratic equation of the form $$ax^2 + bx + c$$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

## Technique

### What is the sum of the possible solutions of $$x^2 - 2x - 3 = 0$$?

\begin{align} x &= \frac{-(-2) \pm \sqrt{(2)^2 - 4(1)(-3)}}{2(1)} \\ x &= 1 \pm 2 \\ x &=3 \text{ or } x=-1 \end{align}

The the answer is $$3-1=2$$. $$_\square$$

The quadratic formula is helpful even when the solutions are complex numbers:

### The roots of $$x^2 - 4x + 5$$ are two complex numbers, $$z_1$$ and $$z_2$$. What is the sum of $$z_1$$ and $$z_2$$?

\begin{align} x &= \frac{-(-4) \pm \sqrt{(-4)^2-4(1)(5)}}{2(1)} \\ x &= \frac{4 \pm \sqrt{-4}}{2} \\ x &= 2 \pm i \end{align}

Thus, the two roots are $$z_1 = 2-i$$ and $$z_2 = 2 + i$$ and their sum is $$(2-i) + (2 + i ) = 4$$. $$_\square$$

Whether the roots are real or complex depends on the quadratic formula's discriminant, $$b^2 - 4ac$$, the expression inside the square root. The roots are real when the discriminant is positive and complex when the discriminant is negative.

## Application and Extensions

### For what value of $$c$$ will $$2x^2 + 7x + c = 0$$ have only a single real root?

The quadratic formula's $$\pm$$ tells us that there will always be two roots unless the discriminant is equal to 0. So,

\begin{align} b^2 - 4ac &= 0\\ (7)^2 - 4(2)c &= 0\\ c &= \tfrac{49}{8} \end{align} _\square

### $$x$$ is a negative number such that $$x^2+9x -22 = 0$$. What is the sum of all possible values of $$y$$ which satisfy the equation $$x = y^2 - 13y + 24$$?

Since $$x^2+9x -22 = 0$$, we know that

\begin{align} x &= \frac{-9 \pm \sqrt{81-4(-22)}}{2} \\ &= \frac{-9 \pm \sqrt{169}}{2} \\ &= \frac{-9 \pm 13}{2} \\ &= 2 \text{ or } -11 \end{align}

Since $$x$$ is negative, $$x=-11$$. So now we need to solve $$-11 = y^2 - 13y + 24$$.

This produces the following quadratic equation:

\begin{align} y^2 - 13y + 35 &=0 \\ (y-5)(y-7)&=0 \end{align}

Thus $$y=5 \text{ or } 7$$, and the answer is 12. $$_\square$$

Note by Arron Kau
4 years, 10 months ago

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