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. $$ 0.75 x^{2}+2.67 x=6.22 x^{2}-4.1 $$

Short answer

The solutions to the equation are \( x = 0.28 \) and \( x = -2.97 \).

Step-by-step solution

Simplify the equation to standard form

Simplify the given equation to the standard form \( ax^{2} + bx + c = 0 \). Start by bringing all the terms to one side, so the equations becomes \( 0.75x^{2} + 2.67x - 6.22x^{2} + 4.1 = 0 \). Then combine like terms, which gives \( -5.47x^{2} + 2.67x + 4.1 = 0 \).

Solve the simplified equation

Use the quadratic formula to solve the equation. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plugging the coefficients into the quadratic formula gives \( x = \frac{-2.67 \pm \sqrt{(2.67)^2 - 4*(-5.47)*4.1}}{2*-5.47} \)

Calculate the solutions

Calculate the values under the square root and simplify the expressions to find the two roots of the equation. The solutions will be \( x = 0.28 \) and \( x = -2.97 \). However, it's crucial to check these solutions by substituting them back into the original equation to see whether they hold true.