A. Introducing Quadratic Equations
Completion requirements
A. Introducing Quadratic Equations
A quadratic equation
is a single-variable polynomial equation of degree two. This means that the equation has a squared variable just like a quadratic function. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
Solving a quadratic equation means determining the value or values of the variable that make the equation true. These solutions are called the roots of an equation. To determine the roots of the quadratic equation, look for values of \(x\) that make the equation true. To aid you in this process, manipulate the quadratic equation until it equals zero. Solving such equations is similar to finding the value(s) of \(x\) that make the function equal to zero \((f(x) = 0)\), or identifying the \(x\)-intercepts of the graph of a given function. Because of your work in Lesson 2.3 on finding the \(x\)-intercepts of the graph of a quadratic function given in factored form, you are already well-equipped to solve quadratic equations.
Note that \(x\)-intercept(s) of the graph of a function, the root(s) of the same function, and the zeros of the corresponding equation have the same value(s).
The function \(f\left( x \right) = x^2 - x - 6\) has \(–2\) and \(3\) as zeros. The graph of the function \(f\left( x \right) = x^2 - x - 6\) has \(x\)-intercepts at \(–2\) and \(3\). This means \(–2\) and \(3\) are the solutions to the quadratic equation \(0 = x^2 - x - 6\).
Recall that the graph of a quadratic function can have \(0\), \(1\), or \(2\) \(x\)-intercepts, which means a quadratic function can have \(0\), \(1\), or \(2\) Real Number zeros. As such, a quadratic equation can have \(0\), \(1\), or \(2\) Real Number solutions.
no \(x\)-intercepts, no Real zeros, and the corresponding quadratic equation has no Real solution
one \(x\)-intercept, one Real zero, and the corresponding quadratic equation has one Real solution
two \(x\)-intercepts, two Real zeros, and the corresponding quadratic equation has two Real solutions
Solving a quadratic equation means determining the value or values of the variable that make the equation true. These solutions are called the roots of an equation. To determine the roots of the quadratic equation, look for values of \(x\) that make the equation true. To aid you in this process, manipulate the quadratic equation until it equals zero. Solving such equations is similar to finding the value(s) of \(x\) that make the function equal to zero \((f(x) = 0)\), or identifying the \(x\)-intercepts of the graph of a given function. Because of your work in Lesson 2.3 on finding the \(x\)-intercepts of the graph of a quadratic function given in factored form, you are already well-equipped to solve quadratic equations.
Recall that to find the \(x\)-intercepts of the graph of a quadratic
function, or the zeros of the quadratic function, you found the \(x\)-values
that made the function equal to zero. By substituting \(0\) for y or \(f(x)\),
you created (and then solved) a quadratic equation. The \(x\)-intercepts of
the graph of a quadratic function and the zeros of a quadratic function
are the solutions to the corresponding quadratic equation.
Note that \(x\)-intercept(s) of the graph of a function, the root(s) of the same function, and the zeros of the corresponding equation have the same value(s).
The function \(f\left( x \right) = x^2 - x - 6\) has \(–2\) and \(3\) as zeros. The graph of the function \(f\left( x \right) = x^2 - x - 6\) has \(x\)-intercepts at \(–2\) and \(3\). This means \(–2\) and \(3\) are the solutions to the quadratic equation \(0 = x^2 - x - 6\).
Recall that the graph of a quadratic function can have \(0\), \(1\), or \(2\) \(x\)-intercepts, which means a quadratic function can have \(0\), \(1\), or \(2\) Real Number zeros. As such, a quadratic equation can have \(0\), \(1\), or \(2\) Real Number solutions.
no \(x\)-intercepts, no Real zeros, and the corresponding quadratic equation has no Real solution
one \(x\)-intercept, one Real zero, and the corresponding quadratic equation has one Real solution
two \(x\)-intercepts, two Real zeros, and the corresponding quadratic equation has two Real solutions