10.9: Summary of Key Concepts
Summary Of Key Concepts
Quadratic Equation
A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a \not= 0\). This form is the standard form of a quadratic equation.
Zero-Factor Property
If two numbers \(a\) and \(b\) are multiplied together and the resulting product is \(0\), then at least one of the numbers must be \(0\).
Solving Quadratic Equations by Factoring
- Set the equation equal to \(0\).
- Factor the quadratic expression.
- By the zero-factor property, at least one of the factors must be zero, so, set each factor equal to zero and solve for the variable.
Extraction of Roots
Quadratic equations of the form \(x^2 - K = 0\) or \(x^2 = K\) can be solved by the method of extraction of roots. We do so by taking both the positive and negative square roots of each side. If \(K\) is a positive real number then \(x = \sqrt{K}, -\sqrt{K}\). If \(K\) is a negative real number, no real number solution exists.
Completing the Square
The quadratic equation \(ax^2 + bx + c = 0\) can be solved by completing the square.
- Write the equation so that the constant term appears on the right side of the equal sign.
- If the leading coefficient is different from \(1\), divide each term of the equation by that coefficient.
- Find one half of the coefficient of the linear term, square it, then add it to both sides of the equation.
- The trinomial on the left side of the equation is now a perfect square trinomial and can be factored as \(()^2\).
- Solve the equation by extraction of roots.
Quadratic Formula
The quadratic equation \(ax^2 + bx + c = 0\) can be solved using the quadratic formula.
\(a\) is the coefficient of \(x^2\).
\(b\) is the coefficient of \(x\).
\(c\) is the constant term.
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Parabola
The graph of a quadratic equation of the form \(y = ax^2 + bx + c\) is a parabola.
Vertex of a Parabola
The high point or low point of a parabola is the vertex of the parabola.