Let g(x)=2 x^2-15 x . Find all values of a for which g(a)=-7...

Name: Problem 70, Chapter 6: Elementary and Intermediate Algebra
Uploaded: 2020-05-26T02:18:25-08:00
Duration: 4 min 29 s
Channel: Melissa Stefan
Description: Let g(x)=2 x^2-15 x . Find all values of a for which g(a)=-7

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two usually expressed in the form ax² + bx + c, where a, b, and c are constants and a is not zero. These functions produce parabolic graphs and are central in algebra, serving as the foundation for many concepts in calculus and applied mathematics.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that satisfy the quadratic equation, typically after equating the quadratic function to a specific value. Common methods include factoring, using the quadratic formula, or completing the square, each of which is crucial for identifying the roots of the equation.

Substitution in Functions

Substitution in functions is the process of replacing the variable with a specific value or another expression to evaluate the function. This technique is often used to set up an equation by equating a function to a given constant, which can then be solved to find the corresponding input values.

Zero Product Property

The zero product property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle is particularly useful in solving factorable quadratic equations, as it allows the equation to be broken down into simpler linear factors that can be individually solved.

Transcript

00:02 We have a function g of x equals 2x squared minus 15x.

00:07 We need to find all of the values for x so that our function is equal to negative 7.

00:13 Let's start with algebraic methods, and then we'll bump over to desmos and see what it looks like on a graph.

00:21 To find all the values of x so that our function is equal to negative 7, we're going to set our given function equal to negative 7.

00:29 Negative 7 equals 2x squared minus 15x.

00:36 In order to solve the polynomial equation, we need to set it equal to 0.

00:40 So let's add 7 to both sides.

00:43 0 equals 2x squared minus 15x plus 7.

00:50 Using the ac method, find ac 2 times 7 is 14.

00:57 And now we need the factor pair of 14 that when added gives us the value of our middle term, which is negative 15.

01:06 In order to have a positive product in a negative sum, both of our factors need to be negative.

01:14 Negative 14 plus negative 1 is negative 15.

01:19 Rewrite the equation so that our middle term is now negative 14x minus x.

01:26 0 equals 2x squared minus 14x minus x plus 7.

01:36 Next we'll go ahead and group.

01:39 Group our first two terms and our last two terms and factor out the greatest common factor for each grouping.

01:47 The greatest common factor of 2x squared and 14x is 2x.

01:52 Dividing each term by 2x leaves us with x minus 7.

01:57 The greatest common factor for x and 7 is 1.

02:02 Negative 1 times x is negative x...

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