Solve each equation using the zero-factor property. -6x² + 7x = -10

Verified step by step guidance

1. First, rewrite the equation so that one side equals zero. Start with the given equation: \(-6x^2 + 7x = -10\). Add 10 to both sides to get: \(-6x^2 + 7x + 10 = 0\).
2. Next, try to factor the quadratic expression \(-6x^2 + 7x + 10\). To make factoring easier, you can factor out a negative sign first: \(-(6x^2 - 7x - 10) = 0\).
3. Now, focus on factoring the quadratic inside the parentheses: \$6x^2 - 7x - 10\(. Look for two numbers that multiply to \)6 \times (-10) = -60\( and add to \)-7$.
4. Once you find those two numbers, use them to split the middle term and factor by grouping. This will give you a product of two binomials.
5. After factoring, apply the zero-factor property which states that if \(AB = 0\), then either \(A = 0\) or \(B = 0\). Set each factor equal to zero and solve for \(x\).

Key Concepts

Zero-Factor Property

The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is essential for solving polynomial equations by factoring, as it allows us to set each factor equal to zero and solve for the variable.

Rearranging Equations to Standard Form

Before applying the zero-factor property, the equation must be rewritten in standard form, meaning all terms are on one side and the equation equals zero. This step is crucial because factoring and applying the zero-factor property require the equation to be set to zero.

Factoring Quadratic Expressions

Factoring involves expressing a quadratic expression as a product of two binomials or other factors. Recognizing common factoring techniques, such as factoring out the greatest common factor or using methods like grouping or the quadratic formula, is key to breaking down the equation for applying the zero-factor property.