Solve using the zero factor property. Be sure each equation...

Solve using the zero factor property. Be sure each equation is in standard form and factor out any common factors before attempting to solve. Check all answers in the original equation.
$$2 x^{2}-4 x-30=0$$

Key Concepts

Solution Verification

After obtaining potential solutions through factoring and applying the zero product property, it is important to substitute these solutions back into the original equation. This step ensures that the solutions satisfy the equation and helps to identify any extraneous solutions.

Zero Product Property

The zero product property states that if a product of factors equals zero, then at least one of the factors must be zero. This concept is applied after factoring a quadratic equation, by setting each factor equal to zero to solve for the unknown variable.

Common Factoring

Extracting a common factor from all terms in an equation simplifies the expression, reducing the complexity of the quadratic. Factoring out a greatest common factor (GCF) is often the first step in simplifying an equation before further factorization is attempted.

Standard Form of a Quadratic Equation

A quadratic equation is typically written as ax^2 + bx + c = 0, where a, b, and c are constants and a is not zero. This form helps isolate all terms on one side of the equation, making it easier to apply factoring methods or the quadratic formula.

Factoring Quadratic Equations

Once in standard form, a quadratic equation can often be factored into a product of two binomials. This process involves finding two numbers that multiply to the product of the coefficient of x^2 and the constant term, while simultaneously adding to the coefficient of the x term.

Transcript

00:01 So let's start with the equation 2x squared minus 4x minus 30 equals 0.

00:08 So we're going to go ahead and solve this by factoring.

00:11 Since it's in standard position, we can go ahead and factor right away.

00:14 And i notice that i have a common factor of 2.

00:17 I can divide each group by 2.

00:19 Take the 2 out and i'm left with x squared minus 2x minus 15, bracket equals 0.

00:26 Now we can go ahead and factor the polynomial inside the brackets.

00:30 Because the coefficient of x squared is now one, we are looking for two numbers that multiply to negative 15, the constant at the end, and two numbers that add to negative 2, the coefficient of x.

00:43 If we take a minute, we can see that we go 5 times 3, we'll get 15, and in order to get negative 2, we need to make sure the 5 is negative.

00:56 So we can go ahead and finish factoring.

00:58 2 bracket x minus 5 bracket and then x plus 3 bracket equals 0.

01:06 Now that we're in a factor we can solve each bracket equal to 0.

01:09 So x minus 5 equals 0, x plus 3 equals 0.

01:14 And we notice the greatest common factor of 2 does not participate in our final answer.

01:19 We're going to solve each of these equations separately.

01:22 So if i add 5 to both sides, i get x equals 5 as one solution...