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. $$ -4.87 x^{2}+1.44 x=5.22 x^{2}+6 x $$

Short answer

The solutions to the equation are \(x = -0.27\) and \(x = 0.27\)

Step-by-step solution

Convert equation to standard quadratic form

Combine like terms on one side of the equation to convert it into the standard quadratic form. Given equation is \(-4.87x^2 + 1.44x = 5.22x^2 + 6x\). Combining like terms, we get \(-4.87x^2 - 5.22x^2 + 1.44x - 6x = 0\). This simplifies to \(-10.09x^2 - 4.56x = 0\).

Apply the quadratic formula

Now we can apply the quadratic formula to solve the equation. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here, \(a = -10.09\), \(b = -4.56\), and \(c = 0\). Plugging these values into the formula, we get \(x = \frac{4.56 \pm \sqrt{(-4.56)^2 - 4*-10.09*0}}{2*-10.09}\). Simplifying, we get \(x = \frac{4.56 \pm \sqrt{20.7936}}{-20.18}\).

Evaluate the solution

Evaluate the expression inside the square root and calculate the two values of \(x\). This results in two possible solutions for \(x\): \(x1 = \frac{4.56 + \sqrt{20.7936}}{-20.18}\) and \(x2 = \frac{4.56 - \sqrt{20.7936}}{-20.18}\). Solving these equations for \(x\), we find that \(x1 = -0.27\) and \(x2 = 0.27\).