Solve each quadratic equation using the zero-factor property. See Example 5. -4x² + x = -3

Verified step by step guidance

1. First, rewrite the equation so that one side equals zero. Start with the given equation: \(-4x^{2} + x = -3\). Add 3 to both sides to get: \(-4x^{2} + x + 3 = 0\).
2. Next, multiply the entire equation by -1 to make the leading coefficient positive, which often makes factoring easier: \$4x^{2} - x - 3 = 0$.
3. Now, factor the quadratic expression \$4x^{2} - x - 3\(. Look for two numbers that multiply to \(4 \times (-3) = -12\) and add to \)-1\( (the coefficient of \)x$).
4. Use these numbers to split the middle term and factor by grouping. This will give you two binomials multiplied together, such as \((ax + b)(cx + d) = 0\).
5. Finally, apply the zero-factor property: set each factor equal to zero, \(ax + b = 0\) and \(cx + d = 0\), then solve each equation for \(x\) to find the solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. It represents a parabola when graphed, and solving it means finding the values of x that satisfy the equation.

Zero-Factor Property

The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is used to solve quadratic equations by factoring them into two binomials set equal to zero.

Rearranging Equations to Standard Form

Before applying the zero-factor property, the quadratic equation must be rearranged into standard form (ax² + bx + c = 0). This involves moving all terms to one side of the equation to set it equal to zero.