Question

square. express each root in simplest 4. $3x^2 + 6x - 24 = 0$

Explanation:

Step1: Simplify the quadratic equation

First, divide the entire equation \(3x^{2}+6x - 24 = 0\) by 3 to simplify it. We get \(x^{2}+2x - 8=0\).

Step2: Factor the quadratic expression

Factor the quadratic \(x^{2}+2x - 8\). We need two numbers that multiply to -8 and add to 2. The numbers are 4 and -2. So, \(x^{2}+2x - 8=(x + 4)(x - 2)=0\).

Step3: Solve for x

Set each factor equal to zero:

• For \(x + 4 = 0\), we get \(x=-4\).
• For \(x - 2 = 0\), we get \(x = 2\).

Alternatively, we can use the quadratic formula. For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions are \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For the original equation \(3x^{2}+6x - 24 = 0\), \(a = 3\), \(b = 6\), \(c=-24\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(6)^{2}-4\times3\times(-24)=36 + 288 = 324\).
Then, \(x=\frac{-6\pm\sqrt{324}}{2\times3}=\frac{-6\pm18}{6}\).

• When we take the plus sign: \(x=\frac{-6 + 18}{6}=\frac{12}{6}=2\).
• When we take the minus sign: \(x=\frac{-6-18}{6}=\frac{-24}{6}=-4\).

Answer:

The roots of the equation \(3x^{2}+6x - 24 = 0\) are \(x = 2\) and \(x=-4\).