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. $$ 3 x^{2}+2 x+5=6 x^{2} $$

Short answer

The solutions to the equation are \(x=-1\) and \(x=5/3\).

Step-by-step solution

Simplify the equation

First, you need to simplify the equation by setting it equal to zero. This is done by subtracting all terms from right side of the equation from left side: \(3x^2+2x+5-6x^2=0\), which simplifies to \(-3x^2+2x+5=0\).

Identify values of a, b and c

Now, you need to identify the values of a, b and c. From the simplified equation \(-3x^2+2x+5=0\), a=-3, b=2, c=5.

Substitute values into the quadratic formula

Insert the values of a, b and c into the quadratic formula. This gives: \(x=\frac{-b±\sqrt{b^2 -4ac}}{2a} = \frac{-2±\sqrt{(2)^2 -4*(-3)*5}}{2*(-3)}\).

Simplify the solution

Upon simplifying the equation, you find: \(x=\frac{-2±\sqrt{4+60}}{-6} = \frac{-2±\sqrt{64}}{-6} = \frac{-2±8}{-6}\). This further simplifies to two solutions: \(x=\frac{-2+8}{-6}=-1\) and \(x=\frac{-2-8}{-6}=5/3\).