QR - Solving Quadratic Functions Lesson

Solving Quadratic Functions

Solving Quadratic Functions by Graphing

graph of parabola with center at (1.5, -12.25) and points plotted at (0, -10), (-2,0), and (5,0) What does it mean to solve a quadratic function? Solving a quadratic function is finding the zero(s), which are the x-intercepts. X-intercepts are the values of x for which the function's value is zero. This concept was discussed in greater depth in previous math courses.

Now we will review how to solve quadratic functions by graphing. The following two videos discuss how to solve, as well as a review of other information we can gather from the graph of a quadratic function. The first video does this without use of a calculator, and the second video solves a quadratic function with the help of a graphing calculator.

Solving Quadratic Functions by Factoring

The first method we will use to solve quadratic functions is factoring. To adequately review this method, we need to review some vocabulary.

A monomial is a polynomial consisting of only one term. A binomial is a polynomial consisting of two terms. A trinomial is a polynomial consisting of three terms. Factoring can be used to write a trinomial as a product of binomials. A quadratic equation in standard form is an equation written in the form equation image indicator LaTeX: ax^2+bx+c $ax^2+bx+c$ , where $LaTeX: a\ne0$ $a\ne0$ . The zeros of a function are the x-intercepts, or the values of x, for which the function's value is zero.

There is a very important property we will use called the Zero Product Property which states that: equation image indicator $LaTeX: \left(x-a\right)\left(x-b\right)=0$ $\left(x-a\right)\left(x-b\right)=0$ , where ab = 0. If ab = 0, then a = 0 or b = 0. This property works only for zero! You can't have two numbers whose product is 5 and assume that one of the numbers is 5 (could be 2.5 times 2)! Again, it's the ZERO product property.

To solve quadratic equations by factoring, we will first set the equation equal to 0, and then factor the equation. Once we have factored the quadratic equation, we will have written the equation in the factored form equation image indicator $LaTeX: \left(x-a\right)\left(x-b\right)=0$ $\left(x-a\right)\left(x-b\right)=0$ . Using the Zero Product Property, we will take each factor and set it equal to zero and solve.

Practically, we need to be able to factor quadratic functions so that we may solve those same equations. You have worked with some methods of factoring polynomials in previous courses. Let us now complete some review of factoring.

Difference of squares only works on binomials with the subtraction operation ("difference" means the use the subtraction operation).
Trinomial factoring is used on any trinomials.

Solve Quadratic Functions by Factoring of the form y=ax²+bx+c, when a≠1.

The following teaching video helps explain very well how to solve Quadratic Functions in standard form y = ax²+bx+c, by factoring when a ≠ 1.

Solve Quadratic by Square Roots

Sometimes when you do not factor, or even when you do factor, your result may be a binomial squared. We can make use of a special method to solve quadratics by square roots. Here are some teaching videos that will walk you through how to use this special method.

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