Question

The solution of a quadratic equation can be found by graphing each side separately and locating the points of intersection. You may wish to consult page 532 for help in approximating solutions. $$ -2 x^{2}+5 x=8 x^{2}-2 $$

Short answer

The solutions to the given quadratic equation are the roots obtained from applying the Quadratic Formula, thus the answer is \(x_{1,2} = \frac{-(-5) \pm \sqrt{(-5)^{2}-4*10*2}}{2*10}\).

Step-by-step solution

Rearrange the Equation

Rearrange the terms to one side to get a standard quadratic equation. In this case, add \(2x^{2}\) to both sides and add 2 to both sides to obtain: \[10x^{2} - 5x + 2 = 0 \]

Apply the Quadratic Formula

The Quadratic Formula is given by:\[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \]For the given quadratic equation \(10x^{2} - 5x + 2 = 0\), \(a = 10\), \(b = -5\), and \(c = 2\). Substitute these into the Quadratic Formula and simplify to find possible values of x.

Calculate Discriminant

The Discriminant \(D\) is \(b^{2}-4ac\). Calculate the value of \(D\). This will determine the nature and number of roots the equation will have. If \(D > 0\), equation will have two distinct roots.

Determine the Roots

Use the Discriminant and Quadratic Formula to find the roots. In this case, calculate the value of roots as:\[x_{1,2} = \frac{-(-5) \pm \sqrt{(-5)^{2}-4*10*2}}{2*10}\]