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Quadratics

A quadratic equation is an equation of the following form: $$ax^2+bx+c = 0$$ where $x$ represents an unknown variable, $a$, $b$, and $c$ are constants, and $a \ne 0$. The left side has all of the fancy numbers and variables, while the right side is $0$.

Because the term $ax^2$ is raised to the second degree, it is called the quadratic term. The $bx$ term is the linear term because it has a degree of $1$. The $c$ term is the constant term because it has a degree of $0$. Quadratic equations can be rearranged to be equal to $0$. Zeros or roots of a quadratic are known as the solutions to quadratic equations.

$a$, $b$, and $c$ don't have to be integers, so you can have a decimal as a constant! Therefore, it may initially seem like you need a perfect square, but you actually don't.

Quadratics can be solved by factorization, completing the square, or using the quadratic formula. To figure out how many solutions a quadratic has, you use the discriminant! You can also find complex solutions with double roots, but that's a little advanced for this lesson!

Now, let's look at some graphs of quadratic functions. Quadratics make a parabola on the graph, which makes a U shape.

Graphed below in red is $y=x^2-2$. We can see that the graph follows the pattern of a traditional quadratic equation, but its minimum point is not on the origin, but $2$ units below the origin.

Graphed below in red is $y=(x-2)^2-2$. Here, we can see that the graph has both been shifted down and also shifted to the right, compared to the original quadratic equation graph.

Graphed below in purple is $y=(x+5)^2-1$. Just like the first two graphs, this purple graph has also been shifted from the original quadratic function graph! Here, the graph is shifted $5$ units to the lift and $1$ unit down.

Graphed below in black is $y=-(x-5)^2+3$. Don't get frightened by this graph! The graph has been flipped because of the negative sign. The graph has also been shifted $5$ units to the right and $3$ units up.

Finally, graphed below in green is $y=x^2$. This is the most basic quadratic function. The minimum point is always on the origin, $(0,0)$.

Looking Further

We will use the concept of substitution later in harder math. So it's best to master it now! Substitution often leads to a quadratic equation when solving systems of nonlinear equations.

Quadratic Equation

A quadratic expression always includes 3 terms: the quadratic term, the linear term, and the constant term. The standard form of a quadratic equation is $ax^2+bx+c = 0$, with a quadratic expression, $ax^2+bx+c$, set equal to $0$. All quadratic equations can be rearranged into standard form.

Quadratic Term

The quadratic term should always be the first term of a quadratic expression in standard form: the term with the $x^2$ in it. So, in $ax^2+bx+c$, $ax^2$ is the quadratic term.

But the last few questions were about an expression, $3x^2+x+4$, not about equations. Let's practice putting quadratic equations in standard form.

All four equations listed are equivalent to the original one, $x = -3x^2-4$, but only $3x^2 + x + 4 = 0$ and $-3x^2 - x - 4 = 0$ are in standard form.

Note: Yes, there's more than one way to write an equation in standard form. There's nothing wrong with that! The only reason we write equations in standard form is so that they're easier to work with in the future. For instance, the quadratic formula is tailored to $ax^2+bx+c=0$ (standard form), not to $ax^2+b+cx=0$ or $ax^2+bx+c = dx^2+ex+f$ (whose formulas would look a bit different).

(Video) Basic Introduction to Quadratic Equations

by Math Booster

This video by Math Booster covers quadratic equations.

$\large{\text{Summary}}$

Quadratic equations take the following forms: $$ax^2+bx+c=y\\ \text{OR}\\ ax^2+bx+c=0$$

where $a, b,$ and $c$ are constants, and $a \neq 0.$ We use the $y$ form for graphing and the $0$ form for solving. The second form is called standard form.

Remember, the standard form for quadratic equations is different from the standard form for linear equations.

There are a few important terms that help us describe the quadratic graph.

The points (or point) where the curve crosses the $x$-axis can be called the solutions, $x$-intercepts, roots, or zeros.

The point where the curve crosses the $y$-axis is called the $y$-intercept.

The minimum is the lowest point on the graph. This will be at the bottom of the curve on a parabola that opens upwards $(a\ge 1)$. Downwards facing parabolas do not have a minimum.

The maximum is the lowest point on the graph. This will be at the top of the curve on a parabola that opens downwards $(a\le -1)$. Upwards facing parabolas do not have a maximum.

The vertex is the point that is the minimum point if the graph opens up. It is the maximum point if the graph opens downwards.

The axis of symmetry is he vertical line that passes through the vertex. It splits the parabola in two equal parts. $x= \frac{-b}{2a}$.

There are many ways to find the solutions to the quadratic equation. We can use factoring, completing the square, graphing, and the quadratic formula. We can also find solutions to quadratic linear systems (which mean multiple equations with at least one linear equation and at least one quadratic equation) using the substitution, elimination, and graphing methods. You will learn all of these in depth in later sections. Quadratic equations are important!