Question

Solve using the quadratic formula: 3x=2x^2-2

Answer 1

Answer:

$x=2\\x=-\frac{1}{2}$

Step-by-step explanation:

1) Move terms to the left side.

$3x=2x^{2} -2\\3x-(2x^{2}-2)=0$

2) Distribute.

$3x-(2x^{2} -2)=0\\3x-2x^{2} +2=0$

3) Rearrange terms.

$3x-2x^{2} +2=0\\-2x^{2} +3x+2=0$

4) Common factor.

$-2x^{2} +3x+2=0\\-(2x^{2} -3x-2)=0$

5) Divide both sides of the equation by the same term.

$-(2x^{2} -3x-2)=0\\2x^{2} -3x-2=0$

6) Use the quadratic formula.

$x=\frac{-b+\sqrt{b^{2}-4ac } }{2a}$

Once in standard form, identify a, b, and c from the original equation and plug them into the quadratic formula.

$2x^{2} -3x-2=0\\a=2\\b=-3\\c=-2$

$x=\frac{-(-3)+\sqrt{(-3)^{2} -4*2(-2)} }{2*2}$

7) Simplify.

Evaluate the exponent

$x=\frac{3+\sqrt{(-3)^{2}-4*2(-2) } }{2*2}$

$x=\frac{3+\sqrt{9-4*2(-2)} }{2*2}$

Multiply the numbers

$x=\frac{3+\sqrt{9-4*2(-2)} }{2*2}$

$x=\frac{3+\sqrt{9+16} }{2*2}$

Add the numbers

$x=\frac{3+\sqrt{9+16} }{2*2}$

$x=\frac{3+\sqrt{25} }{2*2}$

Evaluate the square root

$x=\frac{3+\sqrt{25} }{2*2} \\x=\frac{3+5}{2*2}$

Multiply the numbers

$x=\frac{3+5}{2*2}$

$x=\frac{3+5}{4}$

8) Seperate the equations.

To solve for the unknown variable, separate into two equations: one with a plus and the other with a minus.

$x=\frac{3+5}{4}$

$x=\frac{3-5}{4}$

9) Solve.

Rearrange and isolate the variable to find each solution.

$x=2\\x=-\frac{1}{2}$