Question

8x² - 16x - 42 = 0

Explanation:

Step1: Simplify the equation

Divide the entire equation by 2: \( \frac{8x^2}{2}-\frac{16x}{2}-\frac{42}{2}=0 \), which simplifies to \( 4x^2 - 8x - 21 = 0 \).

Step2: Use the quadratic formula

For a quadratic equation \( ax^2+bx+c = 0 \), the solutions are given by \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 4 \), \( b=-8 \), \( c=-21 \).
First, calculate the discriminant \( \Delta=b^2 - 4ac=(-8)^2-4\times4\times(-21)=64 + 336 = 400 \).
Then, find the roots: \( x=\frac{-(-8)\pm\sqrt{400}}{2\times4}=\frac{8\pm20}{8} \).

Step3: Calculate the two roots

For the plus sign: \( x_1=\frac{8 + 20}{8}=\frac{28}{8}=\frac{7}{2} \).
For the minus sign: \( x_2=\frac{8-20}{8}=\frac{-12}{8}=-\frac{3}{2} \).

Answer:

The solutions are \( x = \frac{7}{2} \) and \( x = -\frac{3}{2} \)