Questions: Solve the equation by using the quadratic formula. 2x(x-2)=5

Solution Steps

The given equation is \(2x(x - 2) = 5\). First, expand the left side:

\[ 2x(x - 2) = 2x^2 - 4x \]

Rearrange the equation to bring all terms to one side:

\[ 2x^2 - 4x - 5 = 0 \]

The quadratic equation is now in the standard form \(ax^2 + bx + c = 0\), where:

• \(a = 2\)
• \(b = -4\)
• \(c = -5\)

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[ \Delta = b^2 - 4ac \]

Substitute the values of \(a\), \(b\), and \(c\):

\[ \Delta = (-4)^2 - 4 \times 2 \times (-5) = 16 + 40 = 56 \]

The solutions for \(x\) are given by the quadratic formula:

\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]

Substitute the values of \(b\), \(\Delta\), and \(a\):

\[ x = \frac{-(-4) \pm \sqrt{56}}{2 \times 2} = \frac{4 \pm \sqrt{56}}{4} \]

Calculate the two possible values for \(x\):

1. \(x_1 = \frac{4 + \sqrt{56}}{4} \approx \frac{4 + 7.4833}{4} = \frac{11.4833}{4} \approx 2.8708\)

2. \(x_2 = \frac{4 - \sqrt{56}}{4} \approx \frac{4 - 7.4833}{4} = \frac{-3.4833}{4} \approx -0.8708\)

Final Answer