Graphing and factoring are just some of the ways to solve quadratic equations. Graphing would not be a very accurate way to solve quadratic equations if the answers are not whole number integers, and quadratic equations cannot always be factored. There are, however, a number of other ways to solve quadratic equations, such as finding square roots and completing the square.
Quadratic Equation: A nonlinear equation that can be written as \(ax2+bx+c = 0\) where \(a \ne 0\).
Perfect Square: A number with a whole-number square root.
Trinomial: A polynomial with three terms.
Quadratic Formula: For \(ax^2+bx+c = 0\), the quadratic formula gives \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\)
Root (of a polynomial): The value that makes the polynomial equal to 0.
Discriminant: The expression inside the square root of the quadratic formula: \(b^2-4ac.\)
In special cases where \(x^2+c = 0\), we can solve the quadratic equation by taking the square root.
For any expression that looks like \(x^2 + bx\), we can always add a constant c so that \(x^2+bx+c\) is a perfect square trinomial. Perfect square trinomial:
To complete the square for \(x^2 + bx\), add \(\Big(\frac {b} {2} \Big)^2\).
These are common mistakes: make sure to avoid them!
Completing the square is useful for rewriting quadratic equations into the form: \(y-k = a(x-h)^2\).
To rewrite \(ax^2+bx+c = y\),
So the vertex is \(\big(- \frac{b}{2a}, c - \frac{b^2}{4a}\big)\)
The quadratic formula comes directly from completing the square. By memorizing the quadratic formula, we do not have to follow the multi-step process of completing the square.
If we know \(ax^2+bx+c = 0\), then we can find \(x\).
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\).
The discriminant can tell us if a quadratic equation will have any solutions before we even begin to solve it. The discriminant is under the square root. We can only take the square root of numbers greater than or equal to \(0\).