Quadratic functions & equations: FAQ

A quadratic is a type of function that can be written in the form $f(x)=ax^2+bx+c$f, left parenthesis, x, right parenthesis, equals, a, x, squared, plus, b, x, plus, c, where $a$a, $b$b, and $c$c are constants.

A parabola is the shape of the graph of a quadratic function. It looks like a symmetrical "U" shape, and can open either up or down depending on the function.

Practice with our Parabolas intro exercise.

Quadratic functions come up in a variety of real-world applications. For example, if we throw a ball into the air, the height of the ball as a function of time can be modeled with a quadratic function.

Practice with our Interpret a quadratic graph exercise.

The quadratic formula is a formula we can use to solve any quadratic equation. It looks like this: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$x, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction, where $a$a, $b$b, and $c$c refer to the coefficients in a quadratic equation written in standard form: $ax^2 + bx + c$a, x, squared, plus, b, x, plus, c.

Practice with our Quadratic formula exercise.

A quadratic equation is in factored form when it is written as a product of two linear factors. For example, $g(x)=(x+2)(x-3)$g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 3, right parenthesis is the factored form of $g(x)=x^2-x-6$g, left parenthesis, x, right parenthesis, equals, x, squared, minus, x, minus, 6. The factored form is particularly useful, because we can set each factor equal to zero to find the $x$x-intercepts of the graph of the function. For example, the graph of $g(x)$g, left parenthesis, x, right parenthesis has $x$x intercepts at $x=-2$x, equals, minus, 2 and $x=3$x, equals, 3.

Practice with our Graph quadratics in factored form exercise.

Vertex form is another way of writing a quadratic equation. It looks like this: $f(x)=a(x-h)^2+k$f, left parenthesis, x, right parenthesis, equals, a, left parenthesis, x, minus, h, right parenthesis, squared, plus, k, where $a$a, $h$h, and $k$k are constants. The vertex form is particularly useful for graphing quadratics because the vertex of the parabola is located at the point $(h,k)$left parenthesis, h, comma, k, right parenthesis.

Practice with our Graph quadratics in vertex form exercise.

Completing the square is a technique we can use to rewrite a quadratic equation or function in a different form. We add and subtract terms to make one side of the equation look like a perfect square. This can help us solve the equation or graph the function.

Practice with our Completing the square (intro) exercise.

Practice with our Solve equations by completing the square exercise.

Different methods work better in different situations. For example, if a quadratic equation is already factored, solving it using factored form will usually be the easiest way to go. But if it's not factored, we might want to use the quadratic formula or complete the square.

Practice with our Strategy in solving quadratics exercise.