Solve the problem using a system of equations. The sum of two numbers is -6 and the product is 8

Solve the problem using a system of equations. The sum of...

Key Concepts

System of Equations

A system of equations is a set of equations involving the same set of unknowns. These equations are solved together because the unknowns are interrelated, and their solutions must satisfy all equations simultaneously.

Substitution Method

The substitution method is a technique for solving systems of equations by expressing one variable in terms of another from one equation, and then substituting this expression into the other equation(s). This helps reduce the system to a single equation with one unknown, making it easier to solve.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two. In the context of problems involving two unknown numbers where the sum and product are given, the formation of a quadratic equation is a common approach, as the solutions to the quadratic can directly provide the values of the unknowns.

Vieta's Formulas

Vieta's formulas establish a relationship between the coefficients of a polynomial and the sum and product of its roots. They are particularly useful in problems where the sum and product of two numbers are provided, as these relationships can be used to construct a quadratic equation whose solutions are the desired numbers.

Transcript

00:01 So this problem it says to use a system of equations to solve the problem, the first thing it says is the sum of two numbers is negative six.

00:19 So sum is negative six, product is, i think it said eight.

00:30 Yeah.

00:31 So you've got two numbers.

00:33 When you add them together, you get negative six.

00:34 So let's just say our two numbers are x and y.

00:38 So we have x plus y is equal to negative six.

00:43 And then their product is 8, so x times y equals 8.

00:52 It doesn't tell you specifically what method to use to solve this.

00:56 I'm going to use simple substitution.

00:59 I'm going to take and solve the first equation for y.

01:09 So if you subtract x for both sides, you're going to get y is equal to minus x minus 6.

01:15 And then i'm going to take and plug that into this equation here for y.

01:21 And so then you would have x times your y is equal to minus x minus six is equal to eight so then x times the negative x would be minus x squared and then x times the negative six would be minus six x equals eight and then we're going to end up needing to factor that and i don't like to have a negative in the front when i try to factor so i'm going to add the x squared over so move that over here and add the six x over so you're going to end up with x squared plus 6x plus 8 is equal to 0.

02:01 Then when you go to factor that, it's going to factor into you're going to have an x.

02:07 And your factors are of 8 are 1 and 8 and 8 and 2 and 4.

02:12 They're going to, because of the positive side in front of the 8, they need to add together to get to your middle number of 6.

02:17 So that leaves the 2 and 4.

02:19 So it's going to be x plus 2, x plus 4.

02:23 So if you follow that, that would be x squared plus 4x plus 2x would be plus 6x, and then 2 times 4 is 8.

02:31 And then you need to set those equal to 0.

02:36 So x plus 2 equals 0, so x would equal negative 2...