Intro to Quadratic equations

A quadratic equation is any equation you can write as $ax^2 + bx + c = 0$, with $a$ not equal to zero.
The number $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant.
You find them by lining your equation up with this general form.

The solutions are the $x$-values where the graph, a parabola, meets the $x$-axis (where $y = 0$).
As the figure shows, a parabola can cross the axis at two points, touch it at one point, or miss it.
So a quadratic has two, one, or no real solutions.

Be careful with signs when reading off $a$, $b$, and $c$.
Rewrite $-x^2 - x + 20$ as $(-1)x^2 + (-1)x + 20$, so $a = -1$, $b = -1$, and $c = 20$.
The number in front of $x^2$ is $a$, even when it is negative.

A solution is any value of $x$ that makes the equation true.
If $x = 4$ is a solution of $x^2 + bx - 24 = 0$, substitute it: $16 + 4b - 24 = 0$, so $b = 2$.
Plugging a known solution back in is a quick way to find a missing constant.

Identify $a$, $b$, and $c$ in $5x^2 - 7x + 2 = 0$.
Compare it to $ax^2 + bx + c = 0$.
So $a = 5$, $b = -7$, and $c = 2$.

Identify $a$, $b$, and $c$ in $-2x^2 + 3x + 1 = 0$.
Rewrite it as $(-2)x^2 + (3)x + 1 = 0$.
So $a = -2$, $b = 3$, and $c = 1$.

If $x = 2$ is a solution of $x^2 + bx - 10 = 0$, find $b$.
Substitute $x = 2$: $4 + 2b - 10 = 0$.
So $2b = 6$ and $b = 3$.