Solve each cubic equation using factoring and the quadratic formula. See Example 7.
$x^{3} + 64 = 0$

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Rewrite the equation $x^3 + 64 = 0$ as $x^3 = -64$ to isolate the cubic term.

Recognize that $64$ is a perfect cube since $64 = 4^3$, so the equation can be seen as $x^3 + 4^3 = 0$.

Use the sum of cubes factoring formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, where $a = x$ and $b = 4$.

Apply the formula to factor the equation as $(x + 4)(x^2 - 4x + 16) = 0$.

Set each factor equal to zero: $x + 4 = 0$ and $x^2 - 4x + 16 = 0$. Solve the linear equation directly and use the quadratic formula for the quadratic equation.

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주요 개념

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Factoring Sum of Cubes

The expression x^3 + 64 is a sum of cubes since 64 = 4^3. The sum of cubes can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). Factoring helps break down the cubic equation into simpler polynomial factors.

Solving Quadratic Equations

After factoring the cubic into a linear and a quadratic factor, the quadratic equation can be solved using the quadratic formula. The quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a finds roots of any quadratic equation ax^2 + bx + c = 0.

Zero Product Property

Once the cubic is factored into products of polynomials, the zero product property states that if a product equals zero, then at least one factor must be zero. This allows setting each factor equal to zero to find all possible solutions.

Solve each cubic equation using factoring and the quadratic formula. See Example 7.x3+64=0x^3 + 64 = 0